You *think* it's "model airplane" glue because that is all you have experience with and assume anything smelling solvents is the same , as thinking all PVA glues is the same.
Sorry, we can't fix your problem.
I would assume that glue sent with a kit, from a reconer supply shop, "should" be suitable.
Yes, solvents in general soften foam and it may twist ... I often use neoprene type contact cement (contains toluene and xylene) and that happens ... in 5 minutes solvents evaporate, foam recovers original shape and adhesive is sticky as it should.
I also use PVA , good quality carpenter's glue, not school kiddie homework stuff but the real thing and it also works very well , both joining foam to paper or polypropylene cone and foam to metal frame, painted or not.
PVA glues are all white while liquid, but the pure stuff dries translucent ; kiddie grade dries white as if it had chalk dust mixed in, because they actually have some industrial grade starch mixed in, for various purposes: thicken/cheapen it, increase "solids content" for peanuts and most important for kids' Moms: it's "washable" from Junior's clothing, so if instead of the cheapest of the cheap you buy Carpenter grade from a paint/hardware shop you'll be fine.