I've had it happen on occasion, in cold, dry conditions...I just ignore it...blood don't really bother me, and I've spilled enough to the road in my time anyhow.
It gets bad enough I just tear some tissues off the minipack in my coat pocket and plug the holes.
As fer lower face protection, I got some bit of skiing gear that covers the lower face quite nicely, even if it does make ya look like hannibal.
Cold weather gear ?
Fullbody snowmobile suit - dubbed "the super suit" cause it allowed me to make rounds at a security job I held in eighteen below once, when no one else would do it...irony of that being, no thief in their right mind would BE out there, but rules are rules.....
Anyhow, layered, thermal undergear, two pairs of pants, sweater.....field jacket liner, armor over that, suit over the works.
Military wool glove liners, and over them heavy cycle gloves with extra insulation that go up to mid forearm and can be zipped over the suit sleeves for full protection.
Boots with two pairs of hunting socks, obviously...once had a battery operated heated pair, but dunno where they got to.
Ski mask, Goggles, lower face protector (ski-wear item) and a Helmet.
Did once have a laffer of a moment at a 7-11 where they called the cops before they realized I was just a half-frozen cyclist looking for some coffee, cause I didn't yank the ski mask before walkin in....even the cops thought that was funny.
Cold ? hah, no such thing as too cold !....