Certified GKH Gardening Expert
If it is that wet it will be very heavy and may be prone to breaking apart at the root ball. Wait until it dries out some more.
If there is limited drainage from the existing pot, you can consider tipping it over on the side (carefully and with some padding to avoid breakage) and drilling more holes in the bottom of the pot.
If it's a ceramic pot you will need a masonry drill bit. If plastic you can use a general purpose drill bit.