Your left speaker is running hot? Rare. Sound Quality OK? Fuzzy sound could indicate a burnt voice coil.
If your speakers are ok and you like them then worth fixing. You can ohm out the voice coils individually as a test of their viability. expect 8 ohms typically printed on speaker magnet and similar speakers should have same resistance.
But if this is a high power bipolar speaker system with a crossover then you may need more expensive caps.
Here you mention a crossover. Speaker cabinets that have more than one speaker such as tweeter and woofer are never built without some kind of crossover.
If your crossover is merely a bipolar cap feeding the tweeter then they truly are cheap speakers as any self respecting crossover has an inductor to limit (choke) the high frequency to the big speaker and save that energy for the tweeter.
middle quality crossover is three way dividing the frequencies up into three bands. Adjustable bands sometimes.
All that said, you could proceed as follows. take the speaker apart and see what you have, Check your voice coils. draw a schematic or download. troubleshoot your crossover and repair or just upgrade your crossover with a good one that you can purchase in kit form.
here is a cheap example I would buy better but you see the cost is not that much.
If you get a good crossover you could use it on your next set of good speakers
oh yea make sure you dont have a dc offset coming from your amplifier