Here's an example of a tamborine playing a different beat than the handclapping done by the choir.

Alcorn Gospel Choir "Lord Help me to Hold Out"

I'm not sure how I would categorize this singing. I think it's an old school gospel song done in a new school gospel way. About 3:30 the choir does some off-beat clapping.

Here's an example of one kind of "contemporary" {new school} Black gospel song:
kirk franklin melodies from heaven

I don't hear any off-beat clapping in the Kirk Franklin song, though off-beat clapping could be done for that song and for any other song.