You may want to look for "gator board" instead of foam core. It's basically foam core with a thin plastic sheet laminated to the sides instead of paper. It's more expensive but comes in a variety of colors, doesn't bend as easily and probably would be safer to paint.
Back in the '50-'70s I built train, slotcar track scenery, buildings, etc., using latex paint until I learned about the special coatings developed for painting flowers. Latex is fine as long as you spray both sides, but the flower paint works great on just one side. Don't recall the brand though. Note that I used an airbrush system for all this except for really large areas, then it was a Binks touch-up gun.
Stop buying using 1$ store Foam board
It's paper is not even glued onto the foam substrate.
Prized By 'frugal' RC airplane types, entirely because the paper literally falls off, allowing access to cheap foam..
Not surprisingly even the foam is low grade... soft /rubbery and of uneven thickness, clearly the disadvantages of 'too cheap'
Suggest buying Decent quality foam core.
Starting with even Michaels cheap "elmers" brand, to Graphic Arts supply pricey/quality Bienfang brand foam core.
Paper won't come off that.. without a chisel
Re Gatorboard: that stuff is really strong and there is even a version with fiberboard sheathing, all are too tough to cut with a razor and require a tablesaw so negates the whole "kitchen table" construction method which makes FC so popular.
Prints are typically Hot press glued to Artists/graphics foamcore..with excellent /durable results.
Spray glued on (3m77) prints wrinkle badly, pretty well immediately on exposure to outdoors humidity. Not a mistake one repeats.
Gator board ( either with plastic or f'glass skins ) is Big for display panel signages and in the film sets industry where it can / is used for 'anything'.
Decent board will not lose it's skin when wet even from Gooped on Latex
IMO Michaels' Elmers' Redi Board brand is as low quality as one can dare use and frankly even that stuff is sketchy. School project quality
all the threads about bracing, materials, damping, multi-layer constructions, translam constructions are flushed down the toilet with the extensive use of FOAMCORE I see lately popping up on diyaudio....
In that case, any thread beginning with the words "mdf or plywood" should be laughed at and locked right away
I had some time over the weekend (snow), so I poured what I had on the small pieces of foamcore, the cheap one from dollar store.
I used gorrila glue, white glue, and mod podge (probably the same as white glue).
Noticable bent is on the white glue, much smaller on the mod podge. If you put just light coat of mod podge, there should be no bent. Perhaps two coats would do it. Gorrila glue does no bent the foamcore, but is pretty unsightly, rough surface, which should not matter if you continue with something to cover it. Mod podge makes nice crusty surface though.