I just ordered the Nelson-Rigg Sierra Dry Adv bags. Haven't used them yet. Like the two other Nelson-Rigg pannier set I had before, all have inner stiffener panels ... so most times inner bags slip in OK.

Do your panniers have inner stiffeners?

My last N/R's had zipper and the whole side of the bag opened all the way up ... so pretty easy to stuff my full inner bag in, which was just a grocery store shopping bag. (Trader Joe's)

But if you OVER LOAD your inner bag with too much CRAP ... then you can't really blame the bags. You're overloaded. Lighten it up and it will fit.

Are you're new GIVI bags are plastic ones? Mine pictured a few posts back.
They're tougher than they look. I fell a couple times, whacked trees and hit a steel pole leaving a parking lot.
A few scuffs, but those GIVI's never cracked. I was impressed how strong.

But you need heavier racks to carry the GIVI's and most ALU panniers. The Jesse bags are great ... but did you ever weigh one empty? no thanks.

Some heavy duty canvas soft bags are heavy, but my Nelson-Rigg aren't too bad, about 3 or 4 lbs. each? vs. 7lbs ea for my old GIVI.

My current racks are super light tubular. 2-3 lbs. ea including steel bolts.

I try to keep all up load about 45 lbs. max. Two panniers and top duffel bag (duffel only half full) Includes clothes, rain gear, tools, bath/first aid, spare tubes, electronics, WATER, extra sandals.