I would suggest running through a small amp and then micing it. It doesn't have to be a small tube amp, solid state will do just fine. If it's small it will probably distort anyway even without built in distortion. That's been my experience. I would suggest finding a way to split the signal (perhaps a small cheap mixer or a guitar pedal with two outputs) and then run a clean signal to the PA or tape recorder, computer, etc. that way you can blend the signal to taste.
Old harp mics look pretty promising too. I've used them before but you may have to try them individually to see how distorted they are. I'm guessing it varies.
I'm a little skeptical of any "mojo" that anyone ascribes to small, cheap tube preamps because I think all of them are "starved plate voltage" designs. Which means there's nothing about them that's similar to any preamps of the sixties. However, that doesn't mean they couldn't be used as an effect if you like the sound of them. The end result is the only thing that matters.