Yes, most banks are failing to do transfers because their 'nostro accounts' are not funded or have very little foreign currency.
There are some banks that will process a transfer for you, so long as you go their with the physical cash for the payment that you want to make. They do not recognise cash in the bank, as they seem to just treat them as numbers.
However, they will only do this for you so long as you are an account holder with them. Ask around from the banks "if I bring you physical cash, how quick will you process my transfer" and go there with the cash when you want to transact.
Let us all know how it goes when you've done it successfully.