I'm in a similar predicament - both my UK bank cards have expired and the replacements get sent to my parents' address, but this year I haven't been able to travel there to pick them up and I don't want to risk getting them sent here via courier. I only use my cards for online transfers, purchases, and payments, which I so far I've been able to do with the new card details, but without the physical cards.
Still not sure what the best option is for sending cash. I use Transferwise for paying other people but you need an IBAN number and DR banks do not have IBAN numbers.