I managed to use every last bit of pulp this time and wound up with 4 sheets of paper. The last sheet was made by pouring the pulp from the vat, through the mold, and into the bucket (pretty messy but easy to prepare for). This last sheet wound up being the thickest of the four and probably the closest to what I'd eventually like to achieve. The third sheet I put on an embossed sheet of felt to see if it'd form to design. Surprisingly enough, it did, though not well enough that it'll come out in the photos.
I let them airdry this time around, at least for a few hours. After that two of them had dried just about completely (though they were a little warped) and I put those two under a couple books. Of the other two I left one out in the open and one under a book.
The following morning I found that the one left on it's own had dried completely but warped in the process. The two that were dry before going into traction were a bit crinkly. The one that was still a bit wet when it went in traction was mostly dry and perfectly flat. So assuming it stays flat going forward I think I have a technique for that.