Cara isn't really a Welsh name (a isn't a Welsh ending) but it 'works' in Welsh.
-an is a common Welsh nn ending, so Betsan's just another nn for
Elizabeth - not as common as
Bethan, but I'd guess that it'll get more popular as
Bethan starts to feel dated. 98 people were given Betsan as a full name between 1984 and 2005, and there were another 6 in 2008. Nicknames as full names are as popular in Wales as they are in England. (woe!)