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...).