If you're worried about small gaps in your construction, use PL premium which is a construction adhesive you should be able to find there at home depot. It is gap filling/bridging to greater than 1/8" while still maintaining strength. The stuff is a bit slippery to work with, but impressively strong once it completely hardens!
Regular wood glue is fine if you've got nice tight joints. I used to caulk all the inside joints as well just for good measure/overkill. My last box I used PL, and had no need to add additional caulking.
If you're going to do it as a furniture piece, being nicely finished/stained or veneered, wood glue is a bit easier to sand flat.
Also, PL is much cheaper than the price you mentioned for flex seal (and you still need to buy glue if you're using flex seal...).
Best,
C.