I have a 2005. The neck dive shouldn't be too bad. But it is noticeable. Here are a few options:
1. Try a leather strap with suede backing and use the suede side against your shirt.
2. Try shortening your strap or lengthening it a bit. I found just last night that raising the bass an inch or so helped to control it.
3. Increase the angle of the neck/fingerboard with the floor.
4. I would ordinarily never recommend this, and it's against my cardinal rules of bass playing, but if you stil can't live with it, try resting your right forearm on the upper bout to counteract the headstock weight.
5. Get lightweight replacement tuners and a heavy brass replacement bridge.
If yours is like most, the next question you'll have is how to get rid of that annoying hum. I just had mine gutted (three of the switches had been damaged during shipping so it was in the shop to begin with). I copper sheilded all of the cavities, brought the body back to my luthier (Jim Mouradian). Jim told me that although there needs to be a good deal of wiring in this bass, all of the wires were about 25%-50% longer than they needed to be. He re-installed the electronics, trimming the wires as needed and soldering grounds to my copper sheilding as needed. The bass is drop-dead silent even with the amp cranked and with Fender nylon tapewounds on it.
Good luck! These basses sound fantastic.