I’m really frustrated because I love to sing, and I can sing in tune just fine, but my tone is simply horrifying. My voice sounds fine to me when I’m singing, but playing back a recording of it, my voice has a stuffed-up, obnoxious quality to it. Is there any way to improve one’s tone? 
I definitely think you can improve your tone. Have you tried voice lessons? Even singing in a choir or something, you’ll probably pick up quite a few things about breathing and raising your soft palate and all the weird things you have to be aware of when you’re singing. I’d do a crappy job of explaining it, but when it sinks in, it’s actually really useful.
The simple answer is yes.
It will take time and work, but you can improve tone.
By the way, how are you recording your voice? If you listen to a recording of even a really good singer on a bad recording it will sound awful. You might want to add some reverb to make your voice sound more natural.
Anyone hearing their voice on a recording for the first time thinks it sounds funny. The resonance in your head gives it a completely different sound from what’s actually out there in the air. It doesn’t matter how good the gear is. Of course, if it’s lousy equipment, that’s even worse.
As others have said, you can improve your tone. Get a good teacher and work at it.
Anyone hearing their voice on a recording for the first time thinks it sounds funny. The resonance in your head gives it a completely different sound from what’s actually out there in the air. It doesn’t matter how good the gear is. Of course, if it’s lousy equipment, that’s even worse.
As others have said, you can improve your tone. Get a good teacher and work at it.