Frankly, unless you find "a very special" Midas shop, that is not the correct answer. ANY of the "big box" stores are notorious for poor depth of service.
I would instead be looking for an independent shop that has been in business for a very long time. You won't find THAT answer on the internet. You'll find it by asking friends, neighbors, by 'asking around.'
YOUR car: Could be a LOT of things, from tires to alignment to worn out parts, and frankly ALL of this
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AS old as this car is now, and IF you have no history of the front suspension ever being rebuilt, I would "just figure" on completely going through the front end, and replace ALL wear parts. Right off the bat
CAVEAT: Replacement parts are getting more and more chineseoated. This is a HUGE problem. Someone over on FABO seems to be having big time problems with Moog ball joints. So as far as replacement brand? "Don't know" anymore. "NOS" (New Old Stock brand name Moog, Spicer, etc)
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Tires. Tires can have a dramatic effect on handling, wander, "cutting" back and forth across the road, but many times, this tire wear has been caused by a front end wear / alignment problem in the first place
Also mismatched tires from front to rear can be a problem. Years ago my boss had a nearly new Dodge pu and needed two new tires. Whatever they were did not like the existing front tires, and at speeds of 50 or more it felt as if it would swap ends, as if wrinkle walls were on the rear!!
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Upgrades. Most of these cars "were minimal" to begin with. Example: My 70 was a "V" code (440 sixpack) and it had the best, the heaviest suspension (track pak) that you could get from the factory. It was "adequate" at the time. But if you bought "that same car" (Belvedere / Satellite) with a 318 or slant six, or even a lower powered 282 Sport Satellite, you may or may not get a front anti-sway bar, you would CERTAINLY get smaller, lesser brakes, and springs and torsion bars which were much much lighter.
So keep THAT in the back of your mind. However, so far as 'going down a straight stretch" of road, even the "lighter" suspension should be able to do THAT
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Alignment. Read the All-par recommendations if you are running radial tires. Recommended settings for radials are different than bias ply. Look down the page for the "skoch chart"
http://www.allpar.com/history/mopar/front-end-alignment.html
That's another issue with front end shops........getting them to vary from old factory bias ply settings.
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WHAT YOU CAN DO
Find a friend who as you describe, is "more" mechanically able. Go to MyMopar and download your shop manual.....free
http://www.mymopar.com/index.php?pid=31
You may have to settle for a Dodge manual over Plymouth, or vice versa. The don't have all of em
Then get you and your friend and go to work. Start "easy." WATCH the steering box and column, while your friend moves the steering wheel back and forth
Examine the column shaft, the coupler, and the shaft into the steering box as it moves......Look for play
Watch the steering box, and look for looseness on the mounting to the frame
Watch the pitman arm as it moves, watching the pitman and the steering shaft. The pitman should have nearly no delay as it moves with the steering shaft
Get "down" on cardboard, etc, so you can see various parts of the "underneath."
Watch the pitman arm and the steering link. The ball connecting them should have NO play as the two of them move together. Ditto each tie rod end and it's corresponding connection.
Ball joints are a little more involved. LOWER balls in these cars are a big time wear problem. Often, when they get bad, they clank and pop. READ the shop manual. You will have to jack the car up under the lower arm, and look for wear.
Upper arm? I would not even try to check it. I would just PLAN on replacing the upper bushings and ball. Likewise, I would plan on replacing ALL rubber bushings........upper arms, lower arms, strut rods, etc.
Work? If you do this yourself it IS hard work. I'm 67, I have severe arthritis, and a hoist. It takes me a LONG time to do this work.
In my "yuth" I could rebuild a front end in a weekend. Front end shops used to HATE that. I'd take the car down, squalling and bawling, squeeling the tires to the shop, with all new parts, and completely out of line. There were two "old guys" (younger 'n I am now) one and the "Chevy" garage and one at the "Poncho" dealer. Those two guys used to align my cars.
Now, in my old age..........I do it myself. I put the car on the trailer, jack it up block it up level, and align it
Bought this toe gauge and scriber off ebay. Had to cut it down, it was too tall (for trucks, LOL)
Bought some old Ammco turn tables cheap
And an old Ammco caster / camber gauge
