Has he used either of those to track down Thomas?
Yes, he used his amulet to find Thomas in White Night after Elaine showed Harry the picture of Thomas walking away with Olivia. Harry even comments that Elaine didn't ask how he was using his amulet to find Thomas.
I know Harry tried to use his blood to track Thomas, but I don't see why he wouldn't have tried to use his amulet first. It would appear to be the simpler method to attempt.
As a side note, I wonder if Elaine eventually put all the facts together. Harry using his amulet to find Thomas and the way they behaved and argued in front of her should have been powerful clues if she thought about it later.