I'll reply here for ya Brody instead of the PM....
first, I would say it sounds like heavy orange peel from the term "goosebumps"...
orange peel can be caused by a variety of things, and combo's of said things..... usually it is caused by thick paint and poor atomization... dry spray can cause it too... a few other more technical things can do it too that i wont get into, cuz i'm not even certain if it's the actual issue...
I'm really shooting in the dark here without some closeup pics of the finish, nor knowing what you sprayed.... I assume it was some sort of rattlecan... enamel, lacquer, i have no idea..
generally the most common issue you'd have would be fisheyeing.... where the paint "walks away" from contaminated areas.. usually these show up as small bullseye type areas.. if the surface has a larger contaminated area, the paint cah actually "sheet" in that area...
pieces like a grill, i'll generally prep them by scrubbing them down with warm soapy water and a scuff pad.. then prior to painting, damp towel/dry towel them with paper towels and denatured alcohol...
the scuffing part is pretty important as it gives the paint a uniform surface to etch to... painting over smooth, shiney surfaces can lead to goofy results.. I usually use red 3M scuff pads, but brillo works in a pinch...
and to answer your sandblasting question.. I'm gonna tell you no... technically it can be done, and i have media blasted plastic in the past, but ya really need to know what to look for when doing it and what media to use.... glassbeading, plastic media, walnuts and sodablasting will give the safest, best results.. standard 0 grit sandblasting would be sketchy and ruin it in a heartbeat without a deft touch... a blast cabinet would be best...
your safest bet would be to let it dry thoroughly then wetsand it with some 220 to 320.. drysanding is not a very good option in such a case.... i can't emphasize enough how important wetsanding is in sooooooo many situations of painting, etc... the only "dry" sandpaper i keep in the house are for my machines, DA, longboard, etc.... ALL the 8" x 11" i use is wetsand...
if the finish is fresh, depending on the paint, you could chemically remove, but obviously this risks melting the plastic if not careful... test spot for that would be in order..
take some closeup pics...