You can find just the wood bead and glue that to the plywood face frames. If you don't want to do true joinery, look up 'pocket screw' to put the frames together. If you can find veneer for the frame edges, that will help keep the edges from sucking up too much paint. Make sure to prime before you paint, I see lots of people try to skip that step and regret it later.
This will take some doing, but it might help. Try a asking a cabinet shop to rip lengths of plywood so all you need is a chop saw. Or see if there is a community college or tech school that will let you use their equipment.