DW Runners

Hi all! I have some questions about sweating/overheating, but since I am new here, first an intro:

I just started running last fall, when I decided to do a Girls on the Run 5k event with my niece. I followed the Couch to 5k plan, and it worked quite well for me, I was able to run 5k by the week before the event. (My focus during the actual Girls on the Run 5k was supporting my niece - she did a lot more walking than running but she finished, felt very accomplished and enjoyed the program overall.) I continued running, with some low points this winter when I wasn't running more than once a week, but recently I've gotten back into a routine of about 3-4 miles, 3 times a week, which makes me happy. I have signed up for a 5k race in a few weeks, and I'm thinking about training to do a 10k by the fall.

But as the weather gets warmer I'm becoming concerned about how to keep cool while running. I sweat very little, and I don't really sweat at all on my face or head. (This isn't new.) Even with temps in the mid 60s I've started to feel uncomfortably hot at times while running. I don't enjoy running indoors on a treadmill, so while I could do that occasionally when the weather is really terrible, it's not a general solution. Ideas so far:

- run early in the morning: not my favorite but I'll do it if necessary
- carry a water bottle: I tried the Nathan QuickShot, which wasn't too annoying to carry. It works fine for sipping water, but that's not my main need; I want to cool my face/head. I found that trying to squeeze water directly on my face while running was only partly successful because the water sprayed all over, got in my eyes, etc. Pouring some water in my hand and then putting it on my face was a bit better. Any other recs or ideas for water containers?
- I have seen cooling bandanas that are meant to be soaked in water and then worn around the neck. Anyone have experience with these?

Any other advice?