Ok, LTC is different then assisted living. There should be more nurses available to orient you. I don't think it should be a scheduling conflict in the sense you are stating.
10 days is not enough time for a proper orientation. However, if that's the most you can get, take it. You are a new graduate for crying out loud! I think you should state this up front. You should not feel ashamed to say it either. The DON should know better.
Let me just tell you this- there have been new nurses at my facility who have gotten very short orientations. Then other new nurses come and they get much longer orientation. The ones who got the crappy orientation will say "what makes her so special that she got so many orientation days". The answere-she asked for them.

If they really want you, they will train you properly. Good luck.
edited to add-congrats on graduating!