Probly not the T/S switch, then
Upper column bearing: grasp steering wheel and attempt to move it up and down and side to side. Any discernible such sideplay = worn upper bearing and/or worn lower bushing in the column.
Steering shaft coupler and steering gearbox: have a helper rock the steering wheel back and forth repeatedly, within its range of easy movement (not trying to actually steer the wheels) while you watch and listen for slop between the steering column shaft and the steering gear input shaft, and evaluate the amount of back-and-forth movement of the steering wheel possible before the front wheels twitch significantly.
There are two adjustments possible on the steering gearbox; see here. The box should also be full of lube, which may very well have all gone away and/or dried out by now. I mix up a delicious recipe of 85w140 gear oil, graphite, and wheel bearing grease.
If you run out of adjustment range before you take out all the slop, the box is worn out and needs rebuilding or replacement. It takes special tools and so isn't usually a DIY project. Interchange is '62-'72 A- and B-body, any engine. V8 boxes have roller bearings instead of bushings, which means less steering effort. If you're buying a rebuild, skip the junk from the local parts store and go directly to a reputable source. Red Head build good boxes (that look nice, too), and they put roller (needle) bearings in place of bushings.
Stock steering gear ratio is 24:1, which means 77.3 (okeh, 5¾) turns lock-to-lock. If you have the money, it is well worthwhile to spend it on a 20:1 box (4½ turns). The steering effort is not appreciably higher, but the quicker ratio makes the car a whole hell of a lot nicer to drive under just about all circumstances. Don't know if Red Head can supply 20:1, but Steer & Gear can (their № 2100), and they also have couplers and rebuild kits, though they list only the '65-up (they say '66-up) style. Not sure what they have for the '62-'64 type coupler; the rebuild kit might work the same, but I just don't remember, nor do I recall if the '65-up coupler is a direct swap in place of the '62-'64 coupler. Firm Feel offer rebuilt boxes in all three ratios* as well as rebuild kits and worm gears in all three ratios if you're feeling ambitious.
*-Do not be tempted by the 16:1 ratio. It is suitable for race cars, but makes for an extremely unpleasant driving (and parking!) experience in a street-driven car, even with an aluminum engine, and especially with heavy things like air conditioner compressors over the front wheels.