Refusing to seek psychotherapy for this situation is like refusing to fix a hole in your tailpipe because you don't trust mechanics. Eventually, it'll rust through and fall off, and you won't have a choice but to fix it... for more money and time.
That is to say, your option is to seek professional help now, while this is a solvable problem, or to wait until things get worse and you don't have a choice. Have you considered what will happen if and when your "meltdowns" escalate to the point where you can't apologize your way out of them? Or, worse, where you are not able to pull yourself out?
You should consider seeking a psychiatrist or physician with experience prescribing psychiatric medication, but be ready for them to refuse to help you unless you commit to talk therapy of some kind also. Be aware that there are dozens of therapeutic modalities that are all different - your previous experience with therapy is unlikely to cover them all. You might have to go through multiple professional people before you find one that will fit what you need.