Turkish Delight is too wet
What you're observing is called syneresis.  Most gelling agents such as gelatin and agar will tend to lose water over time, especially as the temperature goes up (i.e. from refrigerator to room temperature).

What is in fact happening is that the squares are drying out and pushing water out to the surface, which is why the powdered sugar gets soggy or even absorbed (the starch in that powdered sugar keeps it stuck to the jellies).

Gelatin being thermoreversible, you can simply melt it and set it again; it'll be a little on the sweet side due to the increased sugar but otherwise fine.  Of course, you'll just end up with the same results.  You need to add a stabilizer.

Xanthan gum is very good at controlling syneresis and is pretty easy to obtain in health food stores or even supermarkets nowadays.  Add a pinch of that and you'll get much stabler results.  Another additive I've used to stabilize agar-based Turkish Delight is locust bean gum (carob), although I'm not 100% sure if it works as well with gelatin.

Sometimes, you can just let them dry out - periodically wipe the excess water off the surface and hope that it stabilizes.  But be careful; if it's not stable because you didn't use enough gelatin (or didn't use strong enough gelatin) then you might end up with a puddle of goo instead.  So if you plan to try this, monitor it carefully.

Personally, I'd recommend tossing it all in a pot, melting it, throwing in a bit of xanthan gum, and setting it again.  It'll be fine.