Threshold (Target) performance sheets, Better Homes and Gardens (Walmart USA) performance sheets, or Hometrends (Walmart Canada) performance sheets. 400 TC sateen sheets treated for wrinkle-free in 100% cotton. JCP version is called Wrinkle Guard 400TC.
Likely all the same exact manufacturer.
The next step up would be something that is ELS cotton (extra long staple), which would say something like Pima, Supima, Egyptian or Sea Island Cotton.
The elite version of this is Land's End 400 TC No-Iron Supima Sateen Sheets. That's ELS, Sateen, Wrinkle Resistant treated and really LARGE top sheet that will leave plenty of room for the fold over and the tuck-in. No one makes a sheet like Land's End in quality. (Supima is a brand name for ELS/Pima cotton) add a monogram to make them really stand out of top-of-the-line sheets.
Don't buy anything with 600TC or more... it will wear too quickly. Don't buy anything under 160TC as it will feel rough. A sateen weave is better than a percale weave, as it will feel softer. Even Frette sheets, one of the high end brands is just 300TC.
And what you have that has stuff sticking to it... microfiber (or if you prefer the old name, polyester). Bamboo is a Chinese knock-off of rayon, except they don't have trees to make it, so they use bamboo resin, which is a grass but doesn't have the same sheen, so if you are doing Damask stripe, they don't stand out because they just don't have the sheen.
