I have a 1975 Schwinn Le Tour that was given to me so I am not sure if it is the original wheel or not. It is however old enough to use the freewheel setup. The chain and freewheel were replaced about a year ago acording to my friend, however I am having a few issues.

My rear wheel seems to make an awful lot of noise. It's kind of a "clickety clack". It makes the noise whether I pedal or coast but it does seem to be louder when I pedal.

The other issue I seem to be having is more troublesome. My rear wheel on occasion will shift forward and wedge itself against on of the chainstays. It seems to do so more on my left then my right. When this happens I have to loosen quick release readjust the tire and retighten. Sometimes this will fix the issue for a while and sometimes I have to immediatly readjust because it will wedge itself against the stay when I start to pedal. For the most part it seems to be random and usualy happens if I hit a pot hole but it has also happened for what seems like no reason at all. It doesn't seem to matter if I under tighten over tighten or find that sweet spot when locking in the quick release so Im not sure if its me or the wheel.

1. Have the dropout alignment checked to ensure the dropouts are parallel to each other.
2. Make sure the QR lever facing and nut facing are serrated steel - able to engage the dropout facings without bottoming out on the axle itself.
2a. Doesn't hurt to check to see if the axle ends themselves are protruding past the dropouts.

3. Hopefully, the bearings, cones and cup are okay - replace cones and bearings if necessary and re-pack with fresh grease.
3a. If cups are ground out - you're likely out of luck.

4. Also, hopefully your freewheel is not on its last legs.

=8-)

Check to see if your axle is bent....

+1 on the other points, but given the bike and that this may not be the original wheel, this may be the issue. Your bike has dropouts thinner than better bikes, probably about 5mm or so thick. Typical QR hubs have axles that stick 5.5mm into the dropout. So if you put a modern wheel into the frame it's possible that the axle is preventing the QR from clamping against the frame fully.

It's easy enough to check. Leave the wheel in the bike, but unscrew and remove the skewer. Check that the axle is ends at least 1mm short of the dropout's outer faces (both sides).

While you're at it, make sure the QRs conical springs are installed small ends inward -- pointing toward each other like this [>------<], and not like this [<---------->]. conical springs on backward are a common cause of QR issues and wheel slippage.

When I have the wheel off the bike and spin the tire freely or seems to be much quieter with a much smoother sound. I haven't checked the bearings yet but there is plenty of grease coming from the Axel. I don't see too much wear on the freewheel. As far as I'm aware it was replaced a year ago with little to no usage after that.

Among mr. Rabbits suggestions was checking that the dropouts were parallel. That's important, because non parallel dropouts bow the axle effectively tightening the bearings,

If the dropouts are out of alignment is it a fixable issue or do I need to start looking for a new bike? Is this something I can potentially check on my own or is this an lbs issue?

Very fixable. Best done with a pair of tools made for the job. It can also be done yourself, but is a bit more complicated to do. It might be pricy at a bike shop, but hopefully there's a co-op (aka bike kitchen, or community bike shop) where you live, where you can either rent the tools for a low fee, or have them do the job.

BTW- add your city of residency (or a nearby city if you're concerned about privacy) to your profile so folks might direct you to local resources.