Squeezing excess water from seaweed
Most guides I've seen recommend a salad spinner after the initial draining/squeezing. I've never had trouble with that method.

Failing that, I guess you could blot or squeeze it with paper towels; that's what I used to do with lettuce before I had a spinner, and it works... sort of... but not as well, and you'll go through a lot of paper towels that way.

Given that we are talking about seaweed here, I wouldn't try to get it perfectly dry. Restaurants actually use specialized (and very expensive) automatic "vegetable dryers" that I believe operate on a similar concept as a clothes dryer. You're not going to get that kind of result at home. Just squeeze or spin-dry enough to get it past the "sopping" stage.

It's also possible that you soaked it too long. More than a few minutes and it will just turn to mush, which may have been your problem. I've seen at least one ridiculous recipe for seaweed salad calling for a 5-minute rinse and a 15+ minute soak, which is pretty much guaranteed to give you slime.