Joint compound and sanding would restore the walls

what rose said. Drywall finishing is an art. I'd spend a few hundred and get the walls skim coated and finished. then you can prime and paint. However, I'd tile the wall all the way up above the sink. you have the perfect spot for it.

or, finish the wall and paint it a deep color

if you do want to do shiplap, then do the entire wall, all the way up.

Thank you!

Here is a great video I have saved for repairing drywall. I had same problem as you after removing wallpaper that refused to come off. I will need this when on to next paper removal project.

https://youtu.be/_dd9e5pdoD8

Repair the drywall and/or tile. Shiplap has sailed.

Is this a DIY project? If you don't want to tile the space or refinish the drywall ...

https://www.google.com/search?q=white+waterproof+bathroom+wall+paneling&client=firefox-b-1-d&source=lnms&tbm=isch&sa=X&ved=2ahUKEwj2oLXe3JvqAhWhct8KHVxCCzAQ_AUoAnoECA4QBA&biw=1366&bih=654

DO actually look at the panels your local building supply store so you know what you would be getting beforehand. There are a lot of looks/styles, including a ship lap look.

Walls are not ruined. Just skim coat with joint compound and sand. It will be ready for paint (or other finish.)

If you're gong to refinish the walls, invest in an electric sander. Choose the size that will enable you to fold a single 8 1/2" x 11" page of sandpaper twice to create three sections you can cut apart. The older version of this one has lasted well over a decade:

https://www.google.com/search?client=firefox-b-1-d&tbm=shop&q=electric+sanding+tools&tbs=vw:l,mr:1,cat:1188,pdtr0:718548%7C718580&sa=X&ved=0ahUKEwj2-YTG6pvqAhUGU98KHTbgA5AQwQkI7gYoAA&biw=1366&bih=654#spd=10913009109887642682

There are pads for sanding sheet rock but you want the sander that can do more than that and use standard sandpaper for your next project.

Even though this sander is not especially loud and has a dust catcher (remember to empty that), you'll need eye and nose protection. In a small room with trapped sound and/or an echo, you may even want ear plugs.

You have removed the "page" that sealed the sheet rock itself so make sure you choose water proof paint for any painted parts of the walls.

Personally would not crowd this room with tile other than one row over the countertop -- back and sides as a backsplash.

Might consider using white wainscoting on the longer wall.

Make sure your breaker is OFF before you remove the light fixture from the electrical box to redo that wall ... so you will need a drop light and extension cord. Would be a good idea to put tape over outlets so you don't get dust in them.

An electric sander for joint compound would be way too aggressive, IMO.

To avoid the dust from sanding joint compound, I've had good results using a damp sponge to smooth, feather and blend it. Not a difficult DIY task if you take your time with it. I did a 1/2 bath this way after removing the wallpaper caused it to look about like yours.

YMMV

Besides the fact that shiplap is somewhat passe, and wouldn't look good in that space, if you simply add shiplap, it builds out the walls so it would look terrible due to problems with the existing trim, jambs and the crown molding.

If you've never fixed a wall before please invest in hiring a pro to do it. It will not be that expensive. After all the projects I have done I still hire that out. It is an art.