Hello, Pedro!
I've built a few Omni 12's and Omni 10's, and gigged extensively with the O12, less so with the O10. My opinion is that the cabs are intended to be full range instruments, or at least more full range, more truly flat way into the upper registers, than what we as bass players are used to handling. The powerful mids are a good thing for most bassists, the extended high end less of a good thing, as we might start to hear a brittle top end, accented with more fret noise than we want.
The good news is there are easy solutions to this extended fidelity. Rolling back the high end EQ on your bass, your amp, or both will probably get you where you want to be for a nice round, classic R&B tone. It's easier, and usually sounds better, to use EQ to cut overly strong frequencies than to boost frequencies which aren't strong enough.
When I built my first O12, I stuck to the plans and ignored Bill's note that a tweeterless version is bright enough for most bass players. I found that in order to get my sound, I had to roll the highs way back on both my amp and bass. I wasn't comfortable, using most of the available EQ travel to get the high end round enough for my ears, so I installed a fixed 6 dB L-pad (at a cost of about $6) to get the cab's voice tamed a little. The next one will have an adjustable tweeter attenuator, and I would suggest this is a must for most of us, if we build the version with a tweeter.
I think most of the grit in the old school music you're thinking about came from hot tube amps, and maybe even overloaded recording gear, rather than cone breakup. "Cone cry" might be too scary to try to emulate on a regular basis, because it is too easy to destroy the suspension if you overdo it! Using the right amp and gain settings seems like a better way to get the warmth/grit/overdrive that you want to hear.
In short, I think these cabs provide the robust bass and authoritative mid range that you need. The high end that you don't need can be easily dealt with. I do think the cabs need tweeter attenuators, though, if only to provide another tone shaping option. Alternatively, you can build the O12 with no tweeter at all, making it lighter, cheaper, easier to build, and probably better suited to your needs as well!
Good luck!