This sub, a
2002 Velodyne 12" servo sub, I think fits the description of what you are looking for.
It is slightly above your price range, but close. I doubt you will be looking to replace it any time soon. At a buy it now price of $275, I doubt this item will hang around long. I think it is good value.
This certainly will not be a boomer.
By the way changing drivers will not work. The cabinet and porting are specific to the driver.
I was a poor student once, however I still avoided buying junk. I have some equipment I bought when I was a student, still be used regularly on my system now, after a half century. Now that is value.