7 years ago, I picked up a Q*bert turned Arkanoid. The sides had been formica'd over. Underneath was contact cement on the Qbert artwork (badly damaged). Peeled all off sanded, primed.
The I went to Sherwin Williams, got them to color match the yellow out of the sideart (best I could do). In January in my 55 degree garage, I rollered on OIL based paint.
Long story short, after a month, it still was tacky! 2 months more, it was not tacky but was still very soft! Brought it in the house, and finally about 6 months later, it was no longer soft that it kept your finger prints when the sides were pushed hard. A couple years later and it was finally hard.
I thought it was the paint and or coolness. Just a month and a half ago, I took the remaining half quart and rollered it on a PacMan cab that had been painted black. It was 70-80 degrees. Next morning it was still very tacky. Set it out in the sun at about 70 degrees. 6 hours later it wasn't tacky (but picked up a few knats and stuff) but was still very soft.
Repeated it again 2 more days, and finally 3 days later it was only soft if you pressed hard enough around the edges to leave marks. I could finally put it on that side and set it on a blanket to not damage it to finally do the other side.
Fast forward to 2 weeks ago... now it's 50-55 degrees outside, and 60 in my garage. Used RustOleum spray paint and painted another pac cab that had been painted black. (This is all over fully dry Kilz primer on both cabs). Sprayed it, so that I FLOODED the side with about 24 ounces of spray paint. Let it sit for 2 hours, just lightly tacky. Set it up right over night, not tacky in the morning but still soft. Set it outsite in much lower angle sunlight in 50 degree day for 6 hours. When done, paint was hard and I was able to flip it over onto a blanket and paint the other side the same way.
Lesson learned: Don't roll on Sherwin Williams oil based paint... just spend more time priming/sanding the side and just spray it on from rattle cans if you can find a close enough color match.
Now, for latex like you're using, that should get tough enough after a couple days to lay it on something soft like a blanket/comforter.