Learn English – word for losing something in order to gain some other thing

I think the most neutral word is indeed gay, which can be used for both sexes (even though gay woman is not widely used, in favour of lesbian). From Wikipedia:

The most common terms are Gay (both men and women) and Lesbian (women only). [...]

Some organizations (e.g. Safe Schools Coalition) discourage the usage of homosexual in everyday usage, as it might sound too clinical:

Homosexual: Avoid this term; it is clinical, distancing and archaic. Sometimes appropriate in referring to behavior (although same-sex is the preferred adj.). When referring to people, as opposed to behavior, homosexual is considered derogatory and the terms gay and lesbian are preferred, at least in the Northwest.

If the backpack is losing fibers at the edges of the cloth, it is starting to fray.

If entire scraps of woven cloth are coming off, you already know the answer--it's starting to disintegrate.

If it is coming apart at the seams, it is simply coming apart at the seams (an expression that is usually used idiomatically, but which you have the uncommon privilege of being able to use literally here).