Could you hinge them together and make a 'screen' that stands with them? If you stood it with the doors at 90 degree angles to each other, it would be a perfect backdrop for a small table and lamp, etc... They might integrate better in a room that way, as part of a 'tableau'. The 'screen' could also be moved around if wanted and small pictures hung on it still, as you had planned when they were up against the wall.

Many cabs are built this way, but as a DIY project you have to look at how the drawers are built. If the drawer boxes have a front panel other than the actual face, it's pretty easy: I put new fronts on an awful 48inch 'french provincial' [white fiber board] bath vanity a few years back. Between taking the old fronts off [and using them as patterns], building and staining new ones, veneering the face and finishing everything, it probably took me 3 days. Looked a hundred times better and didn't cost much. But removing the old vanity, buying and installing a new one working by myself would have been time consuming, expensive and difficult, so it was a good use of time. OTOH, some drawers will have the face panel as the front of the drawer box, not a separate piece attached to it. On these drawers, substituting finished wood will be a bit tougher. If the drawers and doors are tightly fit, you really don't see the melamine edges until the drawer is opened. Iron on veneer is a realatively easy fix. It doesn't need to be exactly the same species of wood to look good--you'll find oak every where, others range from pretty common to non-existent. You can compensate to some degree by modifying your staining slightly, goes without saying to practice on scraps.

Two things you need to know about solid wood. 1. Wood moves. You cannot stop it but you must allow for it. It is anisotropic. I tell people it's analogous to people putting on weight. If I put on 50 lbs, I would not get much taller. Wood expands (and contracts) minimally in length (along the grain) with absorption (and desorption ) of moisture. It will expand and contract more in two directions, (a) radially, from the center of the tree out and (b) tangentially, along the growth rings, i.e., around the circumference. For most woods, the tangential is approximately double the radial. This is why if you see some dried firewood, you'll usually see a pie-piece shaped crack/void. It's also why plain-sawn wood can cup. If I put on 50 lb, I'd probably put on a significant "beer belly" (front to back) and a little bit less of "muffin top." (side to side) This is why you would never put a solid door inset into a cabinet. In the dry season, it'd have a big gap and in the wet season you may not be able to get it open. As to whether or not it might warp, it depends on how the wood is chosen and what the grain pattern is like. Quarter-sawn wood will have maximal expansion in thickness and minimal in width and less tendency to warp, other things being equal. 2, Wood has planes of weakness and planes of strength. Did you ever see martial arts hit a piece of wood and have it break? Did you ever notice that the grain is running up and down and not between the hands supporting it? Otherwise, they could probably never break it. The wood is weakest up and down the tree. In your second scenario, the grain will be running perpendicular to the rails (horizontal pieces) and if that's only a couple inches wide, it will be very weak. Compound that with the expansion issue above and this is not an option I would ever choose. That is why the "5 piece" AKA frame and panel, is the most common construction method. The rails and stiles have maximum strength and minimal cross grain expansion. If the panel is plywood or some engineered wood, it will have minimal expansion. If it's solid wood (AKA raised panel) then its expansion and contraction is within the stiles' grooves. Solid wood doors often have internal expansion joints and battens on the back to retain them. Engineered woods do not have these issues. Sometimes a sheet, say MDF, is routed out to look like a raised panel, but it's really still just a sheet of MDF. If the corners are rounded and not square, that's the tip-off. https://www.popularwoodworking.com/wp-content/uploads/2010/10/WoodMovement.pdf https://www.thisiscarpentry.com/2010/09/03/moisture-content-wood-movement/

You may never get the new piece to match even if you had the can from the previous finishing. You may need to strip the door and refinish the entire thing to get it to match. Even a few month old finish will have been effected by the sun.