I found a razor just got clogged up too much between the blades.

You might have better luck with one of those bog-standard single-blade safety razors, i.e. one of these:

I've heard from a hairdresser that they prefer the single-blade ones, specifically because the multi-blade ones clog up so much.

I mean, currently I also use a trimmer for my face, as I cannot be asked to do a wet shave every day. But I still use the safety razor for cutting back armpits and groin for hygiene reasons, and it works quite well for that. It's a lot less scary when you don't have to cut thick beard hair.