Another way of doing it is if you've a local veneering firm,get them to glue oak veneer onto a sheet of laminate,you won't need a backing but you will need to hand sand out the initial saw marks on the veneer which won't be a problem as they're reasonably fine.
That way you can cut your MF boards to size,allowing a couple or so mm's for the tops/top rails & bases,& do as others have said regarding keying up the MF face before using contact.Obviously get the veneered laminate made up over size & trim it back to suit.
You can put the unit together as well before glueing the laminate on,then it's a simple matter to fit the frame on the front after trimming back.
That's just a suggestion,it does work & it works well,it's also a good method to use if you want to veneer a curved surface,convex or concave.
Basically it's laminating but with a timber facing.