The alignment shop I brought it to is a one man operation and the owner has been in the business since the '60's. He's built and raced cars locally and currently owns two Mustangs which he's modified to more modern drive trains. Not that any of that means anything, but I feel more comfortable with him than the tire shop that offers free alignments with the purchase of tires. I'm sure they're ok, but I'd bet the alignment guy at the big box tire store had never done a proper alignment on any '60's era cars. For me, I'm more comfortable with the old guy that's been doing it for years.
Sounds like a guy I would trust as well, so the alignment shouldn't be an issue.
This has been my thought for some time. The frame has been patched at some point in time at both rear wheel arches. The face of the left front torque box had pin holes in it five years ago when I bought the car and now I think I can push my finger through it with some effort, the rest "looks fine" from the outside. I put an inspection camera in the frame holes and saw large pieces of rust lying inside the frame, especially the drivers side. I can also stick my finger in the torque boxes and pull out bits of rust. Body mounts are more than likely dried out and cracked.
With all that being said, I found a '68 Galaxie convertible with an absolutely solid frame (no engine or transmission) that I purchased for $1,000. Somehow the car made it up here to NH and was sitting in the back parking lot of an auto repair shop less than 10 miles from me. The body was bad (amazing what Bondo and flat paint will hide) due to Florida salt air but the frame is great. I was able to sell $985 worth of parts that were still usable so I'm into the frame for $15. My plan was to hire a shop to swap out the frame this coming winter but having to replace the roof on my house has delayed my plans. Maybe next year..........
The rust and repairs you found are likely why the handling doesn't feel right.
That left front torque box is certainly an issue.
Rear frame cracking at the wheel arches seems to be a common issue, I've heard of others with those cracks.
I've personally seen 2 Galaxies with cracks in that area.
One a 69 XL and a 66 LTD 2dr Hardtop.
Both belonged to friends of mine and I helped repair the 69 (welding) it lasted about a year and cracked again.
For the 66 LTD, my friend bought a solid 66 500XL with a bad engine (289) and swapped his LTD power train (390) into it.
Glad to hear you have a solid frame to work with.
I repaired mine in this thread...
Edit; I hear you about the roof replacement, I just had mine done, both house and garage, so that expense has cut into my fun car hobby.