Shinybitsofglass, I'm sorry you've joined us. You are going to have to dig deep for the strength to take care of yourself for a bit here. SI is a great resource for doing just that. Read the 180 and start focusing on you, and your health in particular. Think of it as a job - cause it is one of the hardest things to do in the beginning.
Be sure you are getting enough fluids, and eating, you may find you need something to help you sleep. This also means getting tested for STD's. (No matter what he said!)
As for finding out the truth... and because he doesn't seem to want to come clean - you are going to have to dig for the truth (and your sanity). Start thinking like a PI. In short - follow the money, follow the communication, follow the time spent, and if all else fails follow him....
From what you've gleaned so far (Good for you with the bluff btw) I think a couple things can be pieced together. For starters: as of a couple weeks ago he was planning on using your home for his A...
he told me that he needed a huge chunk of time all alone at a certain time every week and that I should take all three kids out someplace so he could have the house to himself.
That's the only thing I can fathom after reading the above.
So either the OW is single and lives at home, married, or she is a professional (in any case he doesn't want to shell out for a motel - or continue to pay for a motel...? Check past credit card bills, atm withdrawals, look for dates that line up with that day of the week.) I suppose it is also possible he is doing something on line and wants privacy... But you need to know what you are looking at. Does he text a lot? look for patterns and that includes regular appearances of gaps, or burst of texting before a gap in communication.
You need to go stealth mode. Is there a password on his phone? Do you know it? Is he often online? Facebook? Just spending time on the computer... (Keylogger if he is.) Where could he have met someone? Work? The gym? The neighborhood? Did he used to talk about a woman from work or where ever, and then stop?
VAR in his car, would likely be telling. Or a gps on the car (especially if he goes out during the hours he requested alone time).
I'm glad you reached out and found us. Whatever happens and whatever you decide - someone here has likely been down that road. It really, really, helps! Hang in there SBoG!