Hello! You are quite right -- an acoustic guitar is not ideal for classical playing. Not only does the skinny neck make it harder to pluck the strings (not enough clearance for your fingers), but the tone is not right: a classical guitar, with its nylon strings and much lighter build, has a beautiful tone that will be very different from what you get with steel strings (a classical will feel nicer to play, as well, with its lower string tension).
There is a section on this forum specifically for advice on buying a guitar, where you may want to post your question (it may get more attention there):
https://classicalguitardelcamp.com/view ... cdd1e8fa05
In your price range, people often recommend Yamaha guitars (but it depends on where you live as to what is available). I've bought guitars online with no problems, as most dealers or even private sellers will pack the guitar very carefully. If you're looking for a trade, perhaps a local shop might be a better option (then you won't have to ship yours). Good luck!