I read that it was illegal to take game fish from one body of water and place them into another.
Bluegill and other sunfish are not game fish. I defer to other posters regarding the water you'd have to use.
I did read on the TPWD website that personally caught bait can only be used on the water (lake/river) where you caught it. Theoretically, they'd like to rule you can't approach a body of water with live bluegill in a container, for instance, because they could assume you are bringing them from elsewhere. However, at the same time, the regulation also says
that you can leave one location (boat ramp) of a body of water and relocate to another location on that same body of water without draining the boat, but that the most direct route must be taken.
The regulation seems terribly written, conflicting, and ambiguous, and certainly a big pain that puts anglers at risk of citations. Again, theoretically by statements from the TPWD website, one should be able to launch at Ramp A, catch bait, reload, then move directly to Ramp B on the same lake, with no requirement to drain the boat. The bait was caught in THAT lake, and you're taking it to another ramp of that lake. You really haven't broken a rule, but I won't be the one to test that - I have bad luck!