I had a pastel portrait of my younger grandson done at Disneyland a little over a year ago. He helped me pick out a frame at the park before we boarded the train for home. Ours was the only roomette with a framed piece of art work in it!
I gave it to his parents, and it was placed in the trunk of my son-in-law's car. Hurricane Harvey visited us a couple of days later, and the portrait remained in the trunk of the car -- which floated a ways down the street in the flood waters -- for several days before the water subsided enough that the car could be cleaned out.
I very slowly and carefully peeled the portrait away from the glass and allowed it to dry. I took it to JoAnn's and worked with a professional framer there. I used a coupon I had gotten in the mail, which brought the cost way down from the original quote. The service was excellent. I got a couple of phone calls during the framing process to keep me informed of the progress and what steps they were taking.
There are some subtle missing pastel spots, and there is a wrinkle across the portrait which is noticeable only if you know it is there. That just adds to the charm of the portrait and makes it even more special. The memories of the trip are wonderful, and just as wonderful are the memories associated with the framed portrait.
I say take your piece of cross-stitch and frame it as elegantly as you can. This is a memory that you do not want to lose. And, by the way, using a larger frame with a thicker matte will raise the glass away from the stitched surface. Your embroidery will be protected from the normal dust and harmful rays of sunlight.