I do it all the time on a tablesaw.
Joint the wood on the face, and an edge.
Trim opposing edge before resawing to ensure it's square.
Adjust blade height to a little over half the height of board. If you encounter resistance either lower height and retry or chuck the board.
Boards will on occasion have internal tension and pinch the blade as you cut. This ...
Adjust your thinking.
You've noted that you're trying to make some trim pieces. I can't, for the life of me, envision what you need 1x1/8" trim for, but I'll assume you need it and that's the perfect size for what you're after. (Oh! You're making doll-houses or other scale models! Of course.)
I'd suggest that you rip your wood in 1/4" wide strips, ...