It is allowed, even if it spills on the tablecloth.
Just be careful to wipe easily and gently to remove the water from there and not to wipe hard or hard to dry it like on a weekday.
Regarding the use of paper for complete wiping, the arbitrators were divided as to whether there is extortion in it and are resorting to stricter due to fear of dauraita and also to use it easily and gently rather than forcefully and urgently, but one can let the tissue paper stay and suck out of itself what it can without the pushing of a human hand.
Sources: Och Shevat and Shem 12 and Shnav letter s.
Regarding paper, see AGM and S. Arhot Shabbat and S. and on Shabbat and Correspondence with Segulah Sheli and more and Achmal.
מק"ט התשובה הוא: 2375