Hi Jake,
Definitely not safe unless the case you use specifically says it's waterproof. A normal case will not provide waterproofing and the cracks in the screen make it unsafe to expose to water.
Unfortunately, even if you get the screen replaced you won't be able to count on the watch being waterproof any longer; any time you remove the screen the IP rating is compromised and can no longer be guaranteed. Your safest bet is to have an Apple authorized service center do the repair, but I suspect they'll tell you the same thing - that it's no longer rated for liquid ingress. Alisha ( @flannelist ), is that your understanding as well?