I treat overshoes as disposable and buy via ebay. Soon as you get in from the ride you need to stuff the shoes with rags or newspaper but keep away from radiators. have a second pair of shoes for the next day. hand dryers good for blowing warm air into the shoes
I know how to dry a pair of shoes, however...
I have two pairs of shoes, but it's no good when the overshoes, socks and shoes are still wet through at the end of the working day, it's 16 miles each way so I'm in the weather long enough for everything to get saturated, especially those Cotswold roads which turn into rivers I've already killed one motor on the e bike from drowning.
The drying room at work is closed due to plague*, and I'm not packing two pairs of shoes to cycle to work am already carrying enough crap as it is.
I've gone through about a dozen pairs of overshoes they never last very long, and no matter what I do the water eventually comes through the cleat holes in the sole of the shoe.
* which I find a space people who ride in once a week like to use as a personal wardrobe and guard space like it's their own. I moved a pair of shoes along to get mine on the drying rack and returned to find my shoes in heap in the corner (and still damp) and the pair I moved a bit back in the middle of a space you could fit two pairs in. This did result in almost everything in the room being ejected by me into the corridor and a very shitty note being stuck to the wall.