I've got a Braun something-or-other (foil), that unfortunately died recently. It has a little plug socket thingy in the base that fitted into the charging station. But it also came with a cable too so you could charge it without using the fancy charging station thing. So I had a shaver, and a cable that could plug into the mains. Question is would the shaver work while plugged in? I don't know because the thing doesn't work any more.
What I'm saying is there may be battery shavers that will work when plugged into the mains, as you require, but don't advertise the fact because people only ever plug them in to charge them, not use them.
If only my Braun was still alive I would be able to check the fact. But it isn't.