What is your vacuum level at idle?
Are you running a brake booster?
Is the brake booster and the engine fighting over the same vacuum at idle?

I agree with a little more advance. Advance it until it pings on acceleration, then back off until the ping goes away.

This sounds like there could be a vacuum leak somewhere. Did it run correctly before and this issue started on its own without having changed anything?

I believe I have an article in the TIPS section of FYIFORD that deals with the Street Avenger carbs. It was posted a few years ago, so you may have to do a search.

15-20K miles should not wear out the throttle shafts.  Plus, Holleys have plastic seals in there.  If you have an excessively strong return spring, well, maybe...

You have opened the idle mixture screws up a bit?

To save you having to dig for it, here is a copy of the Avenger issues I have dealt with in the past.  Pay special attention to #3.There are 3 problems on the Street Avenger that cause the stumbles and hesitations:
1. Lean Jetting. The S/A carbs are jetted very lean on the primary side, and extremely lean on the secondary side. Bump primary jetting up about 3 jet sizes and then make the secondary jets 8 sizes larger than the resulting primaries. To save a few bucks on jets, take the jets out of the secondary side and install them in the primary, and then just buy a pair of jets for the secondary side. This gets the carb jetted about right and will do wonders for throttle response and power.
2. Accelerator pump lever not touching pump cam. I see this constantly on the S/A carbs, and it's really lousy assembly work at the factory: The accel pump lever that connects between the pump cam and the accel pump arm does not ride on the pump cam at idle. The lever is riding on top of the throttle shaft spring and is not touching the pump cam. Only after you rotate the throttle a bit (with no accel pump action) will the pump cam touch the lever and produce a pump shot. To fix this, remove the lever and bend the lever a little more right where you see the "kick-up" bend in the lever where it's supposed to ride on the cam. By giving the lever more of a bend, the lever will not hit the shaft spring and will ride on the pump cam right where it should. This will produce an instant pump shot off idle, and will solve most of the hesitation/stumble issues associated with these carbs. Keep in mind that you will need to re-adjust the accel pump lash adjustment after you bend the arm.
3. Inadequate secondary idle speed setting. On most S/A carbs, the secondary throttle (idle speed) is fully closed. This causes the primary idle speed setting to get cranked up too far in order to achieve a reasonable idle speed, thus exposing too much of the primary transition slot. This, in turn, will cause an off-idle stumble due to lack of transition fuel enrichment as the throttle is opened. The primary and secondary throttles should be open the same amounts at idle - they should be balanced and see equal airflow.  Carb must be inverted in order to change idle speed on the secondaries.

I may have gotten this copy from another source, so I don't take credit for all of it.
 

Opening the secondaries too much will have reduced your idle signal on the primaries and probably worked against you as you observed when you couldn't get it down to idle speed.  You may find that the idle responds better when you back it off to the point where they are just slightly open and then adjust from there in small increments.  Small changes to this setting and then testing takes a little more time but often provides better results.  The system doesn't draw any fuel through the secondaries at idle so if all of the air is going through the secondaries you won't have the fuel to support it.  For the most part, jet size doesn't affect your idle mixture.  Jumping up jet sizes may be helping your performance under load but is likely not going to help your idle/die issue.  The jump you made is quite large so I have concerns that you may be running too rich now.  Did you do this in one change or did you you work your way up to these jets? 

Are you setting the idle screws to 1-1/2 because you think they should be there or are you using a vacuum gauge and adjusting them for maximum vacuum?  Steve suggested opening them up and this is a great test that can give you quick answers.  A change of even a eight or quarter turn can have a large impact on how the engine settles down to idle.  However, an overly rich idle can also cause a stall so you really need to verify with vacuum where you are and what you need to do and the screws may need to go in rather than out.

A bad PCV valve can sometimes produce this symptom.  You can adjust the idle and mixture around it but it will have a potential to die when you let off the gas.

Last, you may be experiencing enough of a voltage drop when you drop down to idle speed to cause it to want to quit due to a weak spark.  Check voltage at the battery at idle and at 2000 RPM and then check voltage at the coil for the same conditions to see if your voltage is dropping due to your alternator or regulator.  Did you replace the pink wire for the Pertronix?  They recommend eliminating the resistor wire for full battery power to the Ignitor.  Some may get away with the pink wire but I tend to follow manufacturer's recommendations.  Also, you have to match the coil resistance to the particular Pertronix unit you are using.  The wrong resistance in the coil may be putting added strain on the Ignitor or your ignition supply wire.  If you are overloading the pink wire it may have slowly degraded over time which could cause a slow decline in your idle spark.  Easy enough to check and since you have not seen any improvement by flogging on the carb, you may want to look to your spark.

It might be helpful to run a long vacuum hose so that you (or better a passenger) can watch vacuum as you drive to see if it is dropping dramatically under certain throttle settings or conditions.

Would a dashpot help?

I agree with Sal. I think you need to advance the timing to about 14 btdc. The way I tune my 289 is to put a vac gage on it and set the timing to achieve max vacuume. Putting a vac gage on it also will give you an indication of a vacuume leak. Also there's always the carb cleaner leak check method. Simple stuff first.

I wanted to give a status update.  A mechanic friend suggested I check my float levels and it turns out he was right.  I had them set way too low and I think the primaries were starving for fuel on hard braking.  I've raised them up and the stalling is gone.  I bought this carburetor new and it came out of the box with see-through sight glasses on the fuel bowls.  They've become cloudy over time and I don't think I was getting an accurate reading looking through them.  I've replaced them with brass plugs that I remove to check float level now.  Thank you to everyone for your input.  I'm so relieved to finally be getting somewhere with this.  I knew it had to be fairly simple but it's been a challenge to narrow down.

eleanor_350 wrote:

[color=#333333]  I've played with float levels, idle mixture screws, secondary idle set screw, timing....nothing seems to help.  .

 
Since changing my float levels when I had the same problem fixed my issue, I'd have offered it as a suggestion if you hadn't included it in your original post   Sorry it took so long to diagnose. What did you do differently this time as opposed to the first go round?

rpm wrote:

Since changing my float levels when I had the same problem fixed my issue, I'd have offered it as a suggestion if you hadn't included it in your original post Sorry it took so long to diagnose. What did you do differently this time as opposed to the first go round?

I wish I knew.  This time I did set the float levels with the site plugs removed.  Before I was trying to set them by looking through discolored "see-through" sight plugs.  Maybe it was distorting my view???  I can't put a big enough disclaimer on the fact that I'm no carb expert.  There was plenty of opportunity along the way for me to think I'm doing right and unwittingly making things worse.

And no apologies necessary.  I appreciate all of the input by everyone.  I'm just glad to finally get it sorted out so that I can move on to other projects.  I self-imposed a requirement on myself to get this fixed before starting any other car-related projects.  Relieved to have this one off my plate.