Long post ahead.
I would actually recommend only using antibacterial soap after training. You didn't say if you were using it before, but don't do that if you are. It will kill the good bacteria on your skin which will actually help keep your skin protected. Defense Soap is awesome! Give that a shot:
http://www.defensesoap.com/
I did start getting a little worse skin from grappling, but it's balanced out. How old are you, if you don't mind me asking? It was worse in my mid 20's, but now that I'm 30, it's not too bad. I always try to get home and shower/wash my face right after class. if you live far away, you could even try to bring face wash to the gym and at least wash your face after class, and I think defense soap also makes some wipes that you can use on your body after class.
I try not to shave my face or neck on days where I do gi. So if I need to shave, I try to wait until there's a nogi day or a day where I won't train. I find the friction of the gi on days where I have shaven already tends to make worse the razor burn and irritated skin. Just doing that alone has helped keep breakouts away.
Another thing I started doing is actually shaving less. I shave maybe once a week, and I just use a beard trimmer to take it really low, and then use a Mach 3 razor on my neck and higher up on my cheeks, so I always have a little but of shadow. I found that wet shaving with a blade regularly made breakouts much much worse. I don't know if you wet shave, or how often if you do, but if you do it often, your hairs grow just a little bit. Then when you shave them back down, they're more likely to start growing sideways and back into the skin. That equals more ingrown hairs/acne/etc. If you have a job that allows it, or if you don't mind how you look with some 5 o'clock shadow I'd advise not shaving as often. I stopped wet shaving my whole face in late 2012 and I've never felt better. If someone clearly has ringworm, impetigo, staph, or something nasty---obviously don't roll with them.
If acne itself is a problem, some local shops can help.
www.downtoearthoils.com has some good products for body and face. Consider seeing a dermatologist and Duac helped me a lot when I was in high school. I went back after doing Jiu-Jitsu and I'd say one tube of it lasts me a year or so. I just use it every now and then if I break out. It doesn't have much Benzoyl Peroxide in it, so you won't accidentally bleach as much of your clothes or pillows, and the other ingredient is Clindamycin. It's not that harsh on the skin and it helped me.
Another great tip is to try to avoid situations where your face isn't getting stuffed into the mat. It happens to everyone, but as you get better you'll be able to avoid that more often.
TL;DR - Don't shave on days where you do gi, consider stop wet shaving frequently, don't use antibacterial soap before class, try Defense Soap, wash face after class at gym, don't roll with people that clearly have ringworm or nasty skin issues, and consider seeing a dermatologist for Duac or something similar. 
