Not necessarily. There are some interesting potential radiation issues with "warp drives" as well. For example, at the boundary of a warp bubble of the generally-Alcubierre-like variety, space is shearing / expanding such that virtual particle pairs are incapable of recombining, just like at an event horizon. That could potentially create a great deal of radiation draining energy from the warp field. Additionally, what happens to particles that are intercepted by the travelling bubble? They don't just disappear!

Wouldn't a jump drive work slightly better in this case?

Meh. We just need radiation shields.