Most cruises that stop at better dive destinations offer a ships dive excursion. Often you can do better by booking yourself with a local operator. The advantage to the ship excursion is that the ship won't sail without you if there's a problem getting back - doesn't happen very often. At many larger cruise destinations such as St. Thomas, Curacao, Grand Cayman, Cozumel etc. there's both a cruise operator and several other options at/near the port.
They work within the cruise timeframes, often offering better dives for lower cost since the cruise dives are often geared around the least experienced cruise diver on the trip (maybe you...

) Often you can get off the ship, do two dives and be back by noon.
A typical example would be Cayman, the cruise operator - Foster's - has several larger boats for their cruise business. Within walking distance are two other 6-8 divers max. operations that may go to better sites. As well as 2-3 shore dive sites. In fairness to Fosters, they maintain the same 8 divers/DM ratio as anybody else on Cayman, just that there may be more groups of them on the boat and at the same dive site. Typically the more people thrashing about in the water, the less marine life. With 16-24 divers per boat and typically 2 ladders, even getting everyone back on board takes time.
We've gotten on boats where there's been 4 of us plus DM and captain, often those turn out to be better/longer dives since at some locations the dive turns when the 1st person goes low on air.
If it's of interest, ask about your itinerary in our cruise forum, someone has likely used all of the port-specific operators previously - and can suggest which locations are better to dive at vs. sightseeing or other activities.
Most LDS's offer trips geared to their customers. Generally they'll have one or several meet/greets with all the trip participants - and accompanying staff member (if there is one) so it's possible you'll have a dive buddy before you go. Most also cost slightly more than if you book the same trip yourselves - often the staff member goes on your dime.
With a non-dive spouse, I'd suggest not going on a dive liveaboard as most are dive focused - you'll do 4-5 dives/day while your hubby mostly sits on the boat - dive sites aren't typically chosen to be near a port or other natural attractions. Aquacat in the Bahamas is one exception, they have kayak tours and several land excursions per week - but even those are geared towards divers looking for something else to do on a non-dive afternoon. Also unless your husband is really interested in diving, he'll be bored to death by all the dive talk, sharing of dive photos, etc. on most liveaboards at night. And you'll be really tired, I've only been on one liveaboard where anyone stayed up much past 10PM - often it's me since I'm an insomniac.