Right now, honestly - good luck finding an apartment in Lake Jackson. With the volume of work going on at Dow, BASF, Quintana/LNG - housing the entire area is at a premium.
I'd look at anything over near the hospital first, then some of the newer apartments down 332 towards Freeport after that. Unfortunately most apartments in the area allow government assistance on rent (even the newer and nicer ones it seems), so you'll always have some questionable renters.