your plant should have bounced back from up potting. there are few issues i see. one, like you said, lack of holes might keeping the soil wet. two, if you added more soil, and if it's not the same soil you used before, the mix of two soil type when not well integrated can cause zone where it will hold more water. last, if it's been watering for weeks and your soil mix doesn't drain well/retain water for long time that's another issue.
i would, as suggested, drill hols first. then i would add a wick just in case so the water will drain faster and quicker without standing water in the bottom. i wouldn't stress the tree by bare rooting it right now unless you are very sure that the soil is just not suitable for the figs in container.
when i bare root the 1 yr old trees, it takes few weeks for them to bounce back. during the time, they look very droopy and some leaves might fall off.