You can hide the fasteners several ways.
How to build simple kitchen cabinets. Overlaps on the sides let you fine tune. The 3 4 plywood case eliminates panel jointing planing and glue ups. You need it to have a case open at the back. If you make sure to countersink your screws you can fill them with wood filler or putty and they will be hidden once painted.
If you have cabinets that you can still make work for your kitchen then by all means keep them and save yourself a lot of money. You can do this by adding inserts that are able to be pulled out. I adjust my face frame to account for. Cut out toe kicks.
This is a comprehensive video that gets into great detail on what is required to make kitchen cabinets including different styles of cabinet face frame and euro style. Add backs to diy kitchen cabinets. This way you have access to water and drainage. I often cover my cabinets with 1 4 beadboard.
Assemble the cabinet carcass. You can cover the sides of your cabinet. Face frames cover plywood edges. Rip plywood down to size.
Add doors drawers. Cut cabinet carcass pieces. However you can still redo them to make them more functional. How to build your own diy kitchen cabinets from only plywood step 1.