Yes, you could change out the plug. What I have done, though, is make up 2' jumper cords with one style receptacle and another style plug to connect my different shop equipment to my outlets. Two jumper cords gets all my equipment plugged in.

There are only 3 wires on my 3 prong plug and existing cord to my welder and now I want to go to a 4 prong but I think I'm missing the white wire on my cord (red, black and green). Will this 4 prong plug still be 50a and safe to use if I don't have a white wire to attached?

Yes, you can connect the new plug as you describe and not have a white wire (neutral). What the neutral wire does is give the appliance the ability to have 120 available to run, say, an internal 120v fan, or a 120v timer on a dryer, etc. Without the neutral wire, the welder manufacturer is telling you that everything inside runs on 240v and therefore does not need a neutral wire. Red and black are the hot wires, green is the ground.

It is possible to make this change, but I don’t see the effectiveness in this. In your case, I advise you to reach out to a specialist. I know what I am talking about since I confronted a similar situation as you. Good thing my brother is a certified welder and was able to show and explain me the situation in my words, so I didn’t have to replace everything by myself and make things worse. He then fixed everything, and I have never had problems since then. That’s when I understood that https://weldingnearyou.com/school-cost/ was a good investment, and the money our family put in there shouldn’t be pitied.