Have you considered anaglypta wallpaper? It can be painted and can look somewhat like moulding.

Here is a link that might be useful: Paintable Wallpaper border

Have you attempted to figure out what textured wall paint was used? Maybe you could match it in texture, remove the border, and then paint with it.

And, a poster here named Funcolors knows of a product that seals over exterior paint and levels it - it is called Peel Bond. She might know whether a similar product could be applied over the interior textured paint. Depending on thw answers you get here, you may want to post on the Paint Forum.

If this were my house, I would be looking at skim coating all the walls and getting them smooth floor to ceiling once and for all.

Why not border over border?

When I moved here 19 years ago, one of the bedrooms had a toy-train border and a toy-train accent wall (other 3 walls were somewhat neutral). Not a great look for a guest bedroom!

I searched for a border that was as big or bigger than the existing border and another accent color that coordinated with the non-toy-train accent wall. I had a professional paperer cover the toy-trains. They used some "special" undercoat. Maybe just sizing. It was totally fine for the duration. I recently took down all of it and painted. Not a big deal and the double layers actually came down easier than the original single layers!

Borders aren't as in now as then, but if you could find a neutral one, it might solve your problem cheaply and easily.

If you don't like the texture itself on the walls, you can probably sand it down easily enough. priming and painting over it may be just fine. It might reduce the texture a little if that is your goal.

I also put a pretty border over a kiddie border years ago...and they're still there!

Your walls are probably deeply textured to cover flaws in the wallboard or plaster. We did that, too, in a bedroom. When we sold the house, the new people asked us how to do it in another bedroom because they liked the look. It was just texture paint.

What about painting the border the same color as the ceiling and doing some kind of trim just along the lower edge of the border? I have seen this done where it makes it look like this substantial crown molding type treatment. I think that the smooth area in contrast with the textured area with the trim material between would make it look even more authentic as if it were expensive molding from the ceiling all the way down the width of the border?