A dryer plug could be a 30a 250v or 50a 250v receptacle. The latte is more common for welders. Check to see how many pins you have and the shape. The receptacle is stamped with the type. Check your breaker panel to see if you have a 30a or 50 amp for the dryer.
Some welders take less than 50 amps and some take more. My tig welder would take 104 amps at 240v single phase. It used much less with three phase and higher voltage, when it was used in a commercial building. I run it on a 50a 250v receptacle and a 50a breaker, but I don't use the notch for 250-460 amps. 5-250 amps is OK depending on how high and how long I run it.
The newer inverter welders take a lot less power and have much better features, and they are smaller the down side is the price. A miller dynasty 200 inverter tig should do what my old transformer based machine will do at 250 amps, but will use much less power. My old machine weighs 1,000 pounds or more and the new one is 30-40 pounds.
My mig welder is a miller 135 110v unit. It does very well for such a small machine. People bag on the 110v machines, but they are great for thinner material maybe up to 3/16" with a single pass. I think the user skill and technique has a lot to do with it.
My brother has a miller 252 mig that will go up to 300 amps, IIRC. I would have to find some very thick metal and up my skills to max that thing out. I wish I had a bigger mig welder, but I would hate to give up the portability of the small one too.
For a 30a circuit you need at least 10 gauge wire for your adapter or extension cord and 50a should be at least 8 gauge. I run a 6 gauge cord on my big welder, but for full output it needs two gauge and should be hard wired. My 110v welder will do ok on a 12 gauge extension cord, if the building wiring is up to it. I ran 125' of 12 gauge for some full output welding to repair a couple gate hinges. The receptacle was in the electrical panel room for the building, so the voltage drop to that point was minimal. It actually welded very well. Other times it sucks on the same long cord. Always go shorter on the power cord when possible, if working toward the limit of the circuit/welder.