The symptoms you describe are not compatible with worn piston rings.  Sounds to me like you have either a bad manifold vacuum leak or a plugged up pilot jet circuit.  

Check for vacuum leaks by spraying down the intake manifold with carb cleaner with the engine idling.  If speed changes dramatically you found your leak.  

Check the pilot jet circuit by tearing down the carb and visually inspecting the pilot jet and pilot passages for crud and blowing through the pilot circuit passages with carb cleaner.  Let us know what you find.

Did you adjust the valves per instructions here or by Clymers?
Clymers has you turn the crank the wrong way and you end up on the wrong TDC.

Your compression test has a lot of variables ,battery charge , if you held the throttle wide open , etc. With your result being 120psi I'd think your compression to be OK.

You have to hold the throttle open when doing the test - or the throttle plate will be restricting the amount of air that reaches the cylinder.

The slide never really closes all the way.  If you look through the carb it stays about 1/4 open at rest. Also, it is not opened by manifold vacuum, it is opened by venturi vacuum which is a function of flow.  The butterfly valve is what controls the air flow.  The slide is used to keep the air velocity as close to constant as possible.  As long as you hold your throttle wide open and override the compression release while cranking you should get an accurate reading.  Your 120 psi seems OK to me.  It wouldn’t require you to keep the enricher on to keep the engine from dying.  The enricher is the dead giveaway.  You either have air leaking in or something preventing fuel from flowing through the low speed & intermediate circuits.

The pilot jet actually has a tapered seat on the top of the jet.  That tapered seat mates with a tapered seat in the body of the carb. When you tighten it up the two tapers lock & seal together.  It should always tighten up and stay put.  Don’t ever put Teflon tape in there.  If it won’t tighten up or won’t stay tight you may have the wrong style jet.  Make sure it has a tapered seat at the top.  Do a Prussian blue check to verify that the two seats are making contact.  If the two seats don’t make good contact air will be drawn in at that point since it is above the fuel level, which will cause a lean condition.

I cleaned the carb out and cleared the Jets to be sure. It started up and was able to idle without the choke this time. I took it for a little drive,  just around the parking lot. While riding I hear a clincking noise coming from the engine. It only happens when it's moving. Not at idle. Could this be an unsuccessful valve adjustment? I just did one prior to cleaning the carb.

I'm with JOG ! If that pulley is not tight it can do some real damage.