Bicycle Tutor

I have '78 Chicago Schwinn Sprint I picked up from Goodwill for $15 and got it overhauled. Anyway, when I shift the rear gears, the shifter slowly moves back up to the highest gear, I could be in the lowest gear in the rear and in 40ft be back up to the highest gear, it shifts through each gear, no skipping. I don't want to keep my hand on the shifter the entire time. If it helps, I got a 77 sun tour rear derailer. Any help will be appreciated.
Thank you,
J

Assuming the cable is tight at the derailleur, what kind of shifter do you have?
Friction or ratchet?
If friction then there is usually some way of tightening it to stop it moving.
If ratchet then possibly the ratchet has seized. You should hear it "click" at each gear position. If not then try some form of lubrication in the shifter. (~ WD40 ).
If you can hear a definite click and the gear still moves back then the shifter may need changing?

It's a friction type shifter, also it is hard to shift it to lower gears, but very easy to go back up to the higher gears.

There should be a bolt where the shifter pivots. You need to tighten that up a little. There's often a wingnut type fitting so you don't need tools. But these were often plastic and may have broken off over the decades. If this doesn't work pretty simply, you might try taking the shifter apart to clean and relube it by unscrewing this bolt. But be very careful to keep track of the order and orientation of each part. There will be several washers that have to go on in exactly the right way.
Stiff shifting may be due to corroded cables. Try dripping a little chain lube on the cable right where they go into the housing.

got it, there is a flathead screw there, tightened that up and haven't had a problem since!