I was told that when the tank is full the gauge reads full but when it reads around half tank it's empty.

So, I took the radiator to the shop today and was told it was not repairable.  I was told that a bad/going bad thermastat could have helped make this happen and after seeing that the cap was bad, blocking flow to the expansion tank, that could also have been a cause by allowing more pressure build up.

Sounds like a bad sender or bad grounding to the tank making the resistance off.

So, the radiator issue is delt with now the cold start problem, If im understanding right the thermowax unit is what closes the choke plate. Since the plate does not close when cold this meens the thermowax unit is bad or could it still be something else? As far as I can tell the thermowax unit is not replaceable so whats the fix?

its been years since I had a sammy with a carb but here is what I remember.  First get a Factory service manual.  you can down load them at some websites. I think suzukiinfo.com.  The carb has three idle settings. First you need to set the warm slow idle-the normal idle.  then you can set the other two.  the cold idle the car must be cold. then the idle up idle for electrical loads.  Plus A/C if you have that. then see it the choke is working.  You have to have the stock choke in CA to pass smog.

Honestly, you will be money ahead to swap to a kick EFI.  I usually find the whole set up here at zuki world for less than the cost of a rebuilt carb, and you probably will never touch it again.

I'm not sure about the thermowax unit being on a carb, I thought that was for the 1990 and newer f.i. rigs. on my Sammie with the stock carb I have allways had to warm it up, shut if off for five minutes or so to get everything under the hood warmed up and then it would run just fine, so maybe it does have some wax in there but I think that the carb could get heat from an exhaust passage if not electrical, I have not messed with mine in a long long time, I might have to see if I can get it running again has been sitting in my barn for a while.

I took this from a cold idle adjustment write up. Is the piece in the middle that the cooler lines go to the thermowax element? Does it control cold idle speed and close and open the choke plate? Also is easily removable?