What do I do? Do I take this new branch off and re plant?? The actual one in soil has turned mushy...

Yes, I would take it off. That bottom part doesn't look healthy at all.

It's one of the ones that the sun shriveled up a couple months back..so far of the 5 only 2 have survived....

I've had rotting issues this year too but largely my fault. I got lazy, bought the wrong potting soil (moisture control) and did not add perlite to the mix. I lost well rooted "red Dauvachelle", JJ's mini-white and JJ's Thumbelina. The bottom part rotted first then it travelled fast to the tip. By the time I noticed the rot, too late to save.

That moisture control soil is really bad for plumeria.

and I know better... I just thought since I am repotting an established tree, that they should be fine with moisture control...👎Wrong!