You've got good timing. I just got mine in the carboy last week too (after a failed batch a few years ago).
I have something that looks like slime too, but I'm fairly certain it's just krausen (which I had less than 5 hours after attaching the airlock).
It sounds like our process was mostly the same (full volume of juice, no campden, pitch and hope it ferments as quickly as possible, so I used a starter with Montrachet), except I added a box of golden raisins to the primary.
One concern I have is you started in a bucket. Usually that's par for the course, but as you mentioned watermelon spoils quickly, so I put everything in a carboy and let it work with an airlock from the get-go.
What's your color like at this point?