You have an AA from 1 CC and 2 (relevant) classes at another CC, yes? How could you possibly only need “a few more classes” to get an engineering degree from UMass? You need 128-136 credits, and at least 45 of them have to be earned at UMass (not transfer credits).
As @Hippobirdy said, you are a transfer student, and they will take credits from as many places as you have them- this is not a novel situation for them.