If I recall correctly, namecheap is just a reseller from enom/tucows. See if you can contact eNom/tucows [1] and explain the situation. Perhaps namecheap is trying to get one last payment out of you? Something sounds off about all this. Maybe conference in namecheap and enom together after talking to enom or have them conference in their people. They probably have contact information that is not publicly available.
If that path is not useful then perhaps pay for the domain so you don't lose it, then have your legal team send them scary letters or take them to court. I am not a lawyer but I have handed off things like this to a companies legal team to sort out for other "registrars".
Yes, namecheap has it as expiring in the dashboard, and I need to pay them to keep it.
The transfer was two weeks ago.
" If still in doubt get on the phone with your new registrar and they can validate if it moved successfully."
I did that before, they said yes, still namecheap in several support mails claimed they have the domain and I need to pay them or lose it.