According to :
Last month we put in a tub surround and used
DAP® ALEX PLUS® Acrylic Latex Caulk Plus Silicone
behind it at the bottom where it meets the tub and around every seam. I
guess this was the wrong caulk.

Now the stuff is just a white goo at the bottom surround seam that
meets the tub and doesn't hold back water. It is a mess.

Interesting. I've found the ALEX PLUS caulk to be the best
value in caulk I've ever used (and I've used _many_) for general
exterior use.

[It's cheap, and is at least as easy to use and durable as any other
except for caulk costing many times as much.]

I've never had it run. But, I've never used on a shower stall/surround/tub.

Or, after I clean up the bottom and the seams, then let this whole mess
dry over the weekend, can I apply the new caulk over the existing
surround seams and bottom that meets the tub?

You should just have to clean it out and redo it with pure silicone.
Like GE-II.
--
Chris Lewis, Una confibula non set est
It's not just anyone who gets a Starship Cruiser class named after them.