Sounds like what you've got there is a classic case of vapor lock. Unfortunately, it's a condition that usually can't be repaired without considerable expense. I'd be willing to trade you a nearly-new Timex for that problem watch to ensure you'd have a fairly accurate timepiece on your person.
Sorry, I couldn't resist. It does seem odd that it will only continue running if the dust cover isn't installed. If it keeps accurate, consistent time when it's working your best bet would be to take it to a watchmaker to be serviced. I'm wondering if snapping the dust cover in place might jar something loose which fouls up the gears, then gets knocked out of the gears when the dust cover is removed. I admit that it's just a guess, though. I think mechanical watches are purty and fascinating, but have no idea how to service them.