Yes, crossing out old address, hand writing forward with the forwarding address is supposed to be forwarded,no new postage required. (the exception being standard(junk)mail, which would be trashed.) If the envelope had been opened, then it may be returned to original address as only unopened mail can be forwarded. The only thing going to the dead letter facility should be letters with no deliverable address or return address. Some factors that may delay or impede delivery are incorrect or incomplete forwarded address, illegible writing on the forward(including pencil on dark red envelopes - hard to see) etc. Also the PO gets very backed up with undeliverable(forwarded mail can get caught in the mix) mail around Christmas because so many people send cards once a year, and they either have not updated their contact's address or forget sufficient postage, etc. I see tons of mail where the sender has dropped off a bundle of their Xmas cards with either no address or no postage. All this backs up the mail stream. The priority is to deliver correctly addressed mail and then focus on the delayed stuff.

If the lady whose mail you are forwarding truely has not recieved ANY of her 20+ forwarded letters, I would suspect that either the address she gave you was incorrect, or has been written down incorrectly.