For years I was (what we call in Washington) a "Married Single Mom" -- meaning that my husband travelled for work far more often than he was home. While my child was a baby and a toddler, all of my friends were busy with careers and none of them had children yet. So, while I can't completely relate, I certainly sympathize with the exhaustion and the lonliness and the yearning to socialize with actual grown-ups...
Here's what kept me halfway sane, for what it's worth:
1) I know that sitters can be expensive, but even if you only hire one for a couple of hours once a month, it's something to look forward to, and I recommend doing this once a week if you can afford it. (babysitting co-ops are a nice option too -- in which you trade "chips" and swap babysitting favors with other families, thus allowing everyone to get out once in a while without spending a fortune. Some churches have these. If not, maybe you could start one?).
2) Join or start a book club, or something that interests you -- and invite friends to meet at your house (after the kids are in bed, or at least fed and jammied, so that you won't have to deal with them, except maybe to tuck them in). Entertaining doesn't have to be stressful or expensive. Your friends will likely be happy with cheese and crackers, cheap wine and good conversation. If you have friends with children, organize a rotating pot luck every other week or once a month. The kids can play outside if the weather permits, or in another room. You can even invite the other children to bring their pajamas, and let them all relax with a movie while the grown-ups socialize.
3)Try to plan a vacation without your children once a year. Seriously, this might be the ultimate sanity-saver. If your parents are inclined to babysit, it's a great opportunity for them to spend quality time with the grandkids. If they can't do this, you might consider a "trade" with close friends. You take their children for a long weekend, for instance, and they do the same for you.
I agree with the poster who suggests that you shouldn't worry much about dating per se. Just get out there and have fun. Socializing with friends is a great way to meet new people (friends of friends) and potential dates.
I hope this helps a little bit...Best luck!