This is the Straight Dope, and by your join date, you may not know how harsh a place this is just yet, but it should be noted that telling dudes to stop by after 10 makes you sound like a Grade A booty call. Maybe even slutty. So yeah, don’t do that. Go on dates in the middle of the day for coffee and you’ll come off as girlfriend material.
As an aside, don’t wait until you’re mid 50’s to get back into the dating game as a single parent. My SO’s mom keeps saying she’s going to try but “one thing or another” comes up, and frankly, she’s just scared to. Single parents will always have excuses, but you owe it to yourself to be happy.
Post on Craigslist, at the grocery store, at Michigan’s student rec center, saying you want $8/hour babysitting. Afternoon babysitting costs less as well. But the best idea so far is do you have friends, or do your kids have friends with single parents? you can both plan a date and share a sitter? Or one watch both kids while the other is out? Seriously get over your fear of having a bad babysitter - I was raised by a wonderful nanny for 7 years of my life, but while she had surgery I had a babysitter who literally didn’t bathe me. So what! I’m alive and no worse for the wear 15 years later! Single parents have this awful “me vs the world” mentality, like people are out to get you.
And don’t meet guys at bars. Jesus, I didn’t think people did this anymore. Statistics show that people most often meet their spouses at work, through friends, and online. Try lots of online dating websites; many are free or low cost. There are some threads here about them.