I can't speak to the BB, but the Fulldrives I have experience with. First, it depends on whether you are talking about the Fulldrive, the Fulldrive 2, or the Fulldrive MOSFET versions. The MOSFET's circuitry can also be found in the overdrive section of the GT-500, another Fulltone pedal.
Usually when people talk about the Fulldrive these days, they are referring to the Fulldrive 2 version, which is the biggest seller. fjs1962 mentioned that he thought the BB was more Marshall-esque, but that it had pronounced low mids. I found that statement a bit confusing to me, since Marshalls are known for having pronounced high mids, versus, say, the Rectos, which are famous for more low mids than are useable sometimes.
The Fulldrive 2 is a very warm sounding pedal, meaning the highs are not particularly overpowering and lending more of a beefier tone to higher notes during solos without sounding harsh or brassy like a Recto on Modern can. I noticed that your signature states that you have a Stilletto Trident, so this shouldn't be a problem for you anyway.
The Fulldrive 2 can also have too much in the low mids if the tone isn't dialed in carefully, though on a Stilletto, again, this won't be as much of a problem. For EL34 amps, the Fulldrive 2 can actually provide a nice balancing tone for those who find the brightness of the EL34's a little too stark. This is especially true for those interested in playing full, rounded blues and jazz tones on their EL34-equipped amp.
Two potential drawbacks of the Fulldrive 2 (as opposed to, say, a Tube Screamer variant) are that: 1) it won't tighten up your bass, and may even add more mud, meaning that really heavy, powerful chugs are harder to achieve with it, and 2) the solo tone can begin to sound a bit "smothered" if you're playing through a darker-sounding guitar like a Les Paul or certain PRS's because of the Fulldrive 2's lack of punch in the mids and high mids, roughly 700 Hz to 3 KHz. It's nothing an EQ can't easily fix.
Overall, the Fulldrive 2 is an extremely warm and natural-sounding overdrive when the gain isn't maxed out. It might just be the perfect solution to someone playing a thin-bodied ash or maple guitar through an EL34-powered amp, where the lows can lack fullness and warmth and the highs can be a bit strident. It also works phenomenally well through the clean channel when set to provide a modest boost with just a smidgen of gain. In fact, the pedal alone when used this way can sometimes produce a better-sounding blues tone than many tube amps' crunch channel set up the same way.
If you are wanting something more typically Marshall-sounding, the OCD is a very nice choice. However, it, too, like the Fulldrives, produces more of a full-range response than any of the more common Tube Screamer clones, so sometimes the bass frequencies of the pedal have to be tamed a bit to avoid mud. Also, be aware that the OCD has enough on-board gain to function both as an overdrive and as a flat-out distortion pedal, so the gain must be used judiciously if you are running it in front of your amp's dirty channels.
