1980 Dodge Other needle valve sticks

The needle may have varnish on it making it stick, especially if the vehicle was sitting for a long period of time with old gas in it. You didn't bother to list the model, engine size, type of carburetor, or any other details, so I can't tell which type of needle and seat you have. Some of them use an "N"-shaped piece of wire between the tab on the float and the needle to help pull it open. Others rely on fuel pressure to push the needle off the seat.

I'd start by replacing the needle and seat. You may have to buy a carburetor rebuild kit to get them but those kits aren't real expensive. If it still won't open, remove the float, lightly push the needle closed, then see it fuel pressure pushes it open when you crank the engine. If it does so poorly or intermittently, check for leaks in the fuel supply line that lets air get drawn in. If the engine runs okay for a few miles before this problem starts, it's almost certain the pickup screen inside the gas tank is plugged or collapsing. Replacements are very inexpensive. The clue is the engine will start and run okay again after waiting a few minutes, but each run will be shorter and shorter before the problem occurs again.

You might try soaking the needle and seat in some type of lacquer thinner or varnish remover if you have some, otherwise the best recommendation would be to just replace them.

The only six cylinder engine used in 1980 was the very tough 225 c.I. Slant six. Most of them came with a Holley single barrel carburetor, but a two barrel Carter carburetor was available too. I own two 1980 Volares with the Holley carburetors, but I never actually worked on one. I've had a number of cars with the Carter two barrel carburetors. They're fairly easy to work on, and you can run the engine with the top cover off so you can watch what the float does and see its setting. The float is held in place with a metal pin about an inch long, and a U-shaped silver metal piece over it. You may need to hold that pin down if the cover is off and the engine is running, otherwise the float might try to float away!