$72K debt is too much, especially when parents haven’t been upfront with OP about how much they will pay, and seemingly don’t have the will or means to pay their $20K EFC. We don’t even know if the parents would qualify for Parent Plus (or other) loans.
Swat is the only school of the 3 that uses CSS Profile, and they came close to the family’s EFC, but did not meet it. That’s why EFCs should be viewed as the minimum amount a family will be expected to pay. Many schools don’t come close to meeting EFC…even meet full need schools, which typically use the Profile, and have idiosyncratic need determination calculations.
As a few posters have noted, OP might look at U Alabama and UAH. Both schools seem to still be accepting apps, but who knows if the colleges of engineering have spots…no harm in calling and asking tho. I expect those schools will cost around the same as LSU, so more expensive that LA Tech. OP, run the NPCs to see what your estimated cost at each would be.