I do the same with gloves although mine are medical grade vinyl type. I do so for two reasons. One is protection of my skin for I do have a sensitive skin. Second is to avoid my contaminated hand from staining/contaminating new paper in box/safe. Although I wash my hands before handling unexposed paper, I'm afraid trace amount or drop on my hand may contaminate it. Now, I just remove my gloves, quick rinse and dry, and I'm ready for the second print.

As to OP's question....

For anything 11x14 and larger and FB, I have problem with getting all surface of paper contacting developer at the same time. This is especially true with FB because they curl toward the emulsion side. I always have some part that refuses to get "wet". To avoid this, I put the paper emulsion side down, slosh, slosh, GENTLY push with my palm, until the paper become saturated then flip it over. Then keep rocking the tray slowly and gently. I dev 2 minutes from the point I flip it over. Usually 2.5 to 3 minutes total. Longer if dealing with warm tone paper. It is REALLY easy for paper to float up and not get enough developer, so I keep watchful eye over it during the whole time.

To dry, I spread bath towel and put my washed and squeegeed prints face UP. Works for me.