A Trend corner chisel will cut 11/32 deep. Might be just right to square a 3/8 groove.
Here's a link. Trend.
Here's a link. Trend.
Not cut a tenon on a table saw but shoulder the end like most mortise and tenon joints do. Then you don't need to worry about the rounded end. The quick illustration I drew shows what I mean. You nibble the bottom front corner of your piece off instead of squaring the end of your groove. This is fast and easy on a table saw by raising the saw blade on it until you have a height that matches the groove depth and running the piece over the saw sideways using the miter gauge with a backing strip attached to it which will prevent the grain from being blown out on the back side. This method is commonly used on things like bookcases where you don't want the grooves that get dadoed into the sides for the shelves from showing at the front. You get nicer cleaner lines on the fronts this way. So you just notch the fronts of the shelves so that the notched out part sits flush against the sides.Thanks everyone for the suggestions. I was hoping to use the router table; probably because I'm new to it and I think it would be "fun" to do it this way. But I think a blind groove (if I'm using the right terminology) that has rounded ends with a square cornered hole/mortise wouldn't look right and I wasn't sure I understood the suggestion to cut the tenon on a table saw - it still wouldn't be round.
I was then wondering if I could use a biscuit bit (don't have tool)...I'm still not sure if that would work. since the wood is only 3/8" and only 2.5" wide.
I tried pre-drilling holes and used small nails and glue - it worked but I don't like it and doesn't feel very strong. I did the same thing with screws. The screws may be the easiest, fastest, all around best way given my tools and wood...but using screws is so boring!
Anyway, thanks everyone for the advice...I'll let you know what I come up with!