I would consider early for TMDSAS as mid July, this is because I would say mid-August is kinda late for AADSAS and considering TMDSAS opens a month earlier than AADSAS, there shouldn't be any reason your application isn't complete.
I would submit your application whenever you feel happy with your essays, the essays are pretty much the only thing adcoms look at to consider interview invites because they don't give out supplementary applications (except Baylor) so you want a good impression and something to capture them right away. Though, if your essays are the absolute last thing missing in your application (given that you already have your DAT score, transcripts, and letter of recs in) I would say turn it in ASAP.