You may need to try another paper. There should be no fading in the water bath following developmen, and only a little during clearing. However, the image should recover all of its density during the first 15-20 seconds in the fixing bath. If you did not tone the image after clearing it may bleach back just a tad during the fixing bath if you fix too long.
Good kallitype papers do not need to be soaked and pressed to work well. If you must go to this much work it would be best to change papers.