There are several schools that offer welding engineering technology, but when I went to school, there were only two in the USA that offered a BS in Welding Engineering. LeTourneau University, Lonview TX, and Ohio State. (Although I heard that Utah State was working on one, but never heard if they completed it.) Also Colorado School of Mines has a good metallurgy program with a lot of welding technology.
Montana Tech in Butte Montana has a great Welding Engineering program. The school is a small engineering school with a total enrolment of about 2500 students. They are currently looking to expand this program to also offer an MS in Welding Engineering.
The welding engineering program is very hands-on as well as technical. Welding Engineers upon graduation can manually weld SMAW, GMAW, FCAW, and GTAW. They also have a robotic welding station. They have an extensive NDE lab where you will learn all NDT processes PT, MT, RT, UT and you will use these processes to qualify all the WPS that are required. Other classes include codes, maintenance welding, and design. The school is very inexpensive when compared to the other two schools mentioned in other post. When I graduated in 1997, I don't think I paid more than $3000 a semester. I was also living in married student housing which cost me $165 per month.
If you are serious, I would urge you to look at Montanan Tech. I believe it’s the best bang for the buck.