I fish out of my own boat in Montauk. I usually use 8 oz jigs and 8-12 oz with a live eel. I've been using 20lb mono, but I plan on spooling with 30lb mono this year. I like graphite, but I really beat up my rods so I should probably stick with fiberglass.
I'm not all that familiar with factory rods, but take a look at some Seekers. They may cost a bit more than you're willing to spend, but they are pretty good rods.
If youre building your own, you can get a Lami CGBT841M. It is a compostie glass/graphite rod, will handle any sinker - this includes 24oz. in a pinch, but is reletively light. This rod will double as a deep water wrechk rod for Blackfish as well, and probably could be used for deep water Cod.
I use an Ugly Stick Model # BWB 1140, Action MH with 20-50 lb line. It's a good stick for the buck (in your price range), and has the potential for blackfish. It's a perfect match for my 310 GTI. It feels real nice in the hands also!
Are you ruling out Penn rods? I have a 330Gti on a penn senator rod. It's an awsome trolling / live bait set up. I found 4fishin.com to have the GTi reels at the best price and they ship fast. The rod I think I got at a sporting goods store like Sports authority or something? The set up is nice! I like the penn slammer and the mariner as well they are cheap and a little lighter action though
My suggestion would be St. Croix. They are some of the nicest rods I have ever used and are more reasonably priced then some of the other great rods, eg: Star, seeker, g loomis, lamiglass. This coupled with a penn international 975, cannot be beat.
Try J&H in Bohemia. I rigged up a penn 975 with a med to heavy shakespere ugly stick. The reel was 189 and the pole was about 55. It is the best setup I ever had. You can use 20-30 line depending on what you want to do.Fast retrieve a smooth as a baby's butt. They have a large selection also for anything you want for tackle. Worth the trip.