I stopped using shaving cream for a while, and tried to get by with just hot water, and my results were markedly negative. Much more irritation, and a lot of ingrown hairs.

I second this. Shave in the shower. I haven't used soap or shaving cream in years. My skin is happier too.

I had the same problem, but it went away immediately after one simple change: stop using shaving cream. Instead, just apply warm water before you shave (it helps to do it after a shower). Before I made the change, my face was always irritable the day of a shave, and exercising would make it flare up; now, nothing. (Having a good multi-blade razor still matters though.)

Maybe you're using the wrong cream? Using just water was much worse for me than using cream. (I've also found than massaging for a while to spread the cream rather than just sticking it on my face helps.)