BAD Splash Paint. HELP!

Personally I think texturing would look like just exactly what it was- covering up a problem.

Take a stiff putty knife and scrape off what you can. Then do some more. The more you get off this way , the easier the next part. Do not gouge the sheetrock- that causes more problems.

Using premixed topping like Plus 3 , and a wide mud blade -12" ( also have a 6" to get it onto the big one and help to work it) skim what unevenness is there.
Let dry. Sand smooth ( use a light to the side when sanding to see)
Prime mud- look over again- spot what ever needs it, resand those and spot prime, paint 2 full coats.

This is done all the time to fix surface problems.

Good luck!

I would also use some sanding cloth after scrapping with the putty knife, instead of sand paper. It's sold in the drywall area. It's not going to load up like sand paper. Then follow the other directions.
May be good enough to get out of skim coating.

But if you really want a knock down texture remember texture will hide a multitude of sins. Meaning if your texturing it don't have to be that good. Smooth walls yes don't skip a step.