You don't indicate what is your connecting flight. From your profile, I assume that it is GOT but it might help to know.
I side with Often1 (although it seems he mistook TG for TK).
Whose fault is irrelevant at this stage. TG is notorious for its poor IT.
You did not make things easy by calling AF US (e.g. DL) and having a ticket reissued on DL stock.
IMO doing a three-way call will be hard to achieve. Going to the airport early seems the best option, BUT I believe that DL has no operations in BKK anymore and AF only two flights per week (Thu and Sat), so you will not get them at the airport.
Buying your own ticket could turn out very expensive for you.