I think your situation is analogous to a first date: are you obligated to share every part of your private history on a first date? No. It's just a first date, not a marriage proposal.
I think the over weighted sense of an obligation to tell the "truth" also comes from a place of shame. You are entitled to your privacy. If you want to keep the appts, then let them know you have pre-scheduled appts that would not be possible to reschedule at short notice. I don't think you need be any more specific than that.
Any other concerns are way down the road at this point, and those bridges can be crossed when and if you reach them.