Pretty good so far. One thing you can do is use a slow curing glues as Charles suggested, or one of the brown glues that take hours to cure completely.
I suggest you use a small metal brush Pix below, to apply the glue. These give you pretty good control of the glue. Generally you can wipe off any excess using a damp cloth for a couple of passes on the inside and outside. If you made the fingers a little proud (desirable), you will be using a block plane or sanding to get rid of the excess.
The other challenge is to keep the box square as the glue sets. If the bottom fits tight, you can insert it to help hold it square. Another option is using box joint cauls (pix) and a belt style band clamp (Pix). The cauls stand clear of the slightly proud fingers of your box joints. The picture of the orange band clamp already has a standoff and should work well
You could make your own L shaped cauls on a table saw, adding a strip to each inside edge so they make room for the proud fingers. The L drawing is an end view. Start with a 2 inch square piece. Make certain your saw blade is exactly 90 to the table. Make one cut about 1 3/8ths high, flip the piece and cut the same amount on the other side. Use some aged and seasoned wood so it will stay stable. Glue some strips on the inside edges as shown so they stand clear of the glue. Use a band clamp as shown to hold the box together. Cut the cauls into about 6 inch long sectons and wax the holy crap into the inside so the glue doesn't stick!
You could also use some reliably 90 degree L brackets, clamped with small clamps, into each inside corner. Make sure the brackets don't touch the glue. The second diagram shows how simple it is to make someting to hold corners square on a table saw, using 2 inch square material. Cut square by having the blade be exactly 90 to the table, then make a 45 degree cut as shown. I would use some 4 inch F clamps--five bucks each at Home Depot--see pix. It will take at least 8 clamps for a box like yours. You could use the small Harbor Freight F clamps for this instead, but I like the Bessey clamps better.
Finally, what you have made is a box joint. Finger joints are different. Each finger tapers and they are usually used to glue two short pieces together end to end to produce longer lengths. Many inexpensive items are made from this kind of joint. Because finger joints are tapered and come to a point, they cannot be used for box joints. Just a little woodworking lingo. And here are a couple of links. https://www.amazon.com/RAM-PRO-Flexi...gateway&sr=8-3 https://www.rockler.com/box-joint-cauls https://www.amazon.com/DCT-Multi-Ang...30-spons&psc=1
There are a lot of ways to glue up box joint boxes, these represent a few that will do the trick for you.