What does the welding schools locator at this site turn up near your home?
Are you planning to go full time for 4 years or spread it out with work study?
One path might be to get a degree in manufacturing or mechanical engineering with an emphasis on welding. Or, you could complete as many core courses as possible locally and then transfer to another school. But, most schools won't take more than 4 semesters in transfer credits. It pays to plan ahead to avoid extra efforts.
And AWS currently accepts experience and other degrees for their certified welding engineer exam. They will even take a high school diploma with the right experience. But, it is a very hard test. I have looked at some of the sample questions. It is no walk in the park.
I have never done a detailed statistical study to see what percentage of employers will accept a BSME vs. a BSWE for welding engineering positions. But, I did just read that 75% of BSWT grads are able to hold down full engineering positions. That has been true of my career path so far. But, a BSWE will go farther with the government and certain heavy R&D jobs or for a PE exam.
Oh, and I earned a second BS in Computer Science at the age of 37. So, don't let age scare you. There were people older than me in some of my classes. I was near a navy base and there were a lot of retired navy people going to school.