Try using nonstandard razors.

I tried a traditional straight-edge - my experience is that it's too much bloody work just to keep the thing in working order. Also, a wet strop is a ruined strop, in an environment in which it's likely to get wet - not workable if you share bathroom space with other people who may not be as careful of your stuff as you are. Or animals who think it makes a nice chew/scratching toy.

However, you can find straight edge razors which accept single-edge razor blades (the cheap replaceable kind), and also safety razors in either the double-edge or the straight-edge variety.

Straight edge with replaceable blade advantages:

They don't get clogged.

Substantially cheaper to use.

Closer shave.

You don't need to trim a beard before you shave it off.

Disadvantages:

Until you figure them out, and probably even after you do, you'll cut the heck out of yourself.

Unwieldy for some applications (upper lip, I'm looking at you)

Unusable for some applications

You might get mistaken for a hipster or contrarian or both.

Safety razor with replaceable blade advantages:

Cheaper

Substantially harder to clog, and easily remedied - pull the blade out, rinse it off, put it back

Closer shave than the X-blade monstrosities

Substantially more stylish than plastic

Disadvantages:

Sharper, AKA "Safety" is relative - unlike the more conventional plastic things, I find shaving cream of some kind to be an actual necessity, and irritation seems more frequent.