Chockostang  to get yours rebuilt.
He may have rebuilt boxes available too.

Talk to Bearing Bob about his 19:1 box first.

I removed mine while rebuilding/modifying front end.
I went through it and removed as much of the old grease and repacked with new.
Word of caution, don’t loose any of your bearings in lower section.
They are not caged.

This car had power steering from factory as well.
I made a slight adjustment on box and it reduced the lock to lock turns as well.
This adjustment should be done without anything connected on output shaft.
The Haynes manual had the procedure in it...a bit confusing at first.
Actually made more sense after disassembly.

BTW, throughly clean exterior of box before opening it up.
I used a brass wire brush and degreaser.
I’m glad I did.

I used a 19:1 Chocko rebuilt manual box in my 69 for less than 1,000 miles. While they're nice in parking lots, I don't recommend using one if most of your driving is not in parking lots (car parks for our British folks). I'll make someone a smoking deal on it if they need it.

The REAL GT350 Shelby used the same box but had longer pitman and idler arms which further reduced lock-lock steering travel.  Opentracker sells the kit.

SHELBY QUICK STEER KIT

PREPARE for a lot of exertion to corner and park the car.  You ain’t getting any younger, BILLY!

Ditto on chockostang box rebuilds. He gets good ratings. 

Or just but a Borgeson p/s box and be done with it.

Grease would certainly help. I tore down 3 Mustang boxes in the last two years and none of them had anything resembling grease in them. They all had a small amount of hard black chunks that at some time was probably grease. The best looking one from the exterior had mucho rust damage to the sector shaft and worm gear which made then useless. I won't use a fifty year old steering box that hasn't been rebuilt.