If they aren't nursing, they'll leak, so that's not the issue IMO. Spoken as the father of 3 breast fed babies, not from a deer physiological standpoint. But I know from experience that they don't just fill up and explode, they fill up until the "pressure relief valve" gives and they leak. Not that it can't lead to this, but I'd be more inclined to think this is the issue.
Having deer in the city where I live, they run around here like pets. I’ve seen that pretty frequently to the point it seemed normal. Sat 10 feet away from fawns nursing on a mother that looked like that. I’m no expert though.