I was trying to be humorous I guess. So is it your suggestion that the scope should stay? Nobody has really stated what they would do in my shoes. Again, I'm rather new to this, and appreciate the experience of others more familiar......
My 2c: when I look at those maximum-vertical-zoom, non-bandwidth-limited, no-averaging, no-hi-res screenshots, they
both look terrible. Superimpose a signal of interest on top of that, and it's going to be a horrible experience trying to sort the signal from the noise. This isn't a fault with the scope, you've just put it in the worst case position. If you look at how the noise is 1.5mV or whatever at 300 MHz, that's extraordinary performance for the price point (maybe, I dunno). Set it back to 1V/div, and it'll look perfect.
What you're doing is taking a race car, driving it around an oily hairpin bend at 300 miles per hour and complaining that the tyres have slightly different lack of grip while you smash into the railings. Turn on averaging or hi res, orset it to 1 V/div, and forget about this "problem". Is my very long 2c.
If your scope has significantly (2x) worse noise than another copy of the same model, then that's a different story.