For you who have played the scramble, you might enjoy this variation called the shamble. In the Golf shamble format, things change after everyone tees off. When you're playing a scramble, every player on your team takes a whack at the ball from the tee and then on from where the team determines is the "best" lie for the next shot. In the shamble, everyone tees off as you do in the scramble. Just as in the scramble, everyone on the team plays from what is determined to be the best shot. Here's where the similarity ends, though. In the shamble, from the second shot to holing out, everyone plays their own ball. So you have a mix of features from the scramble and from stroke play. It's nice to have four shots (shambles are typically in teams of four) from the tee and be able to pick the best. From there, though, the pressure's on because you pick the best score amongst the four.
Because of the mix of abilities typical in such tournament formats, it's reasonable to use players' full handicaps. Of course you can add all sorts of variations onto the shamble as is the case with scrambles. You might use two or three of the best scores rather than just one. If you wanted to add a bit more pressure to the teams, you could require up to four drives from each player in a foursome be used.