I'm no pro, but I'll offer a few suggestions...
First... It's a great song and you have the voice to pull it off. I totally agree with the comment above that a full band version sould wound awesome. It depends on what you're after, I guess. Slightly overdriven vocal,
acoustic guitar only (and the shaker, which is cool), and ultradry mix with kind of an aggressive song and performance style is kind of a cool effect. It doesn't by any means sound bad. If it's what you're after, you can make it sound better with an unclipped recording of the vocal and a different means of creating the overdrive. Another treatment (short of of full band production), though, would be to put a little 'verb on it! I don't know whether Audacity has gotten any better since I last used it (maybe 5-6 years ago), but I would recommend giving Reaper a try. I'm not promoting for them or anything, but if you're not in place where you can drop a few hundred bucks on software (which I'm guessing you're not if you're using Audacity), I think Reaper is haaarrrd to beat! You can import the tracks you already have. To those you could add a teeny bit of slapback (ReaDelay), and some reverb (Reaper has two modules that are pretty good) to all your tracks. You have to be careful not to overdo it. What I try to do (when not trying to make a noticible echo effect ) is just add some ambience to the track without making it muddy and without making it sound like it was recorded in a well or a bathroom. It's the only way I've found to mostly make up for not having a good sounding room. You record everything close enough to the mic to be loud enough that you don't pick up much room sound (careful not to clip/pop). Raw, the tracks will sound really dry and close. Then you add a little (teeny bit) of slap back and barely audible reverb (and usually a little bit of compression on vocals) to try to make it sound like the track was recorded in a good sounding room with the mic back from the source a bit like I imagine the pros did in the old days (I've read that even professionals nowadays use close micing in a dead room and add room sound with effects, but I don't know that to be true.) Anyway, once that is done, I think of that as my "dry" track for mixing, and add "effects" from there as desired.
The other thing Reaper has that helps me A LOOOOT is a pitch correction plug in. Your track is quite good, but there are a few spots that a wee bit off. I know some folks have an attitude about pitch correction, but unless you're Whitney Houston, I think you're gonna need it to one degree of another. EVERYTHING you hear on the radio nowadays is pitch corrected - EVERYTHING! So when you record something that has even small little pitch bobbles, it really jumps out. The best thing is that when you have only a few small bobbles, which is all I heard in your song, you can correct it without affecting the sound of your voice AT ALL!
Like a say, I'm no pro, but I've been messing around with home recording for a long time (starting with ping-ponging between two cassette tape decks in high school.) I just got my first decent mic for Christmas, and I have to admit that has made a noticable difference. Even before that, though, I think some of the stuff I've recorded has sounded really good. It's more about figuring out how to make what you have sound as good as it can. Most important thing, to me, is to get the levels right when you record the track. That absolutely can not be fixed after the fact. Just right is best, but quiet/noisy is better than overdriven/clipped. That's important to remember mainly if you're recording somebody else. Try telling you primadonna lead singer that he has to sing his perfect take over because you had the level too high (really because he sang louder and closer, but that doesn't matter now does it?) and it's all crackly and sh*t!!! Been there, done that, not fun!! If its your own take, well... you only have yourself to blame.
Crap I've written a book. Sorry about that. Anyway. Good song. Keep at it. Don't worry about equipment...
J