Morinja, you risk smothering the roots if you attempt to replant when the soil is soaked. If you press down on the new soil and push out all the oxygen from the rootball, you could have problems.* Been there, done that!* I* would suggest you wait until the soil dries out a bit if you want to repot it..* If it's a little rootbound and you can take the entire rootball out of the pot to let it air dry without the rootball falling apart, give that a go.* Don't put it in full sun out of the pot for too long as you risk any exposed roots getting sunburned.*
If you find the soil falls away from the roots when you take it out of the pot, you could put the tree back in the pot with new soil, but don't pack the soil down until the rootball dries out some.* The new soil might even help to wick away some of the moisture.*
I don't remember how big your tree is or when you last repotted it, but another solution might be to unpot the tree, rinse off all soil and repot with fresh soil.
When weeding, the best way to make sure you are removing a weed and not a valuable plant is to pull on it. If it comes out of the ground easily, it is a valuable plant.