what are some of the ways to dive warm waters (and see more than sea grass) and spend less?
While there's ways to economize in Cayman, it's basically known for being an expensive place overall. (That does come with plenty of nice restaurants, accomodations, lots of good dive ops with nice boats/facilities, few bugs, and a very 1st world experience.) It's a nice place to start - though new divers that are sticking shallower at first may miss some of the wall stuff it's famous for. Food is expensive, accomodations can be expensive, diving will add up. You can save in Cayman by finding less expensive accomodations like condos with kitchen facilities so you're not eating out all the time, maybe not on the water. It's a great place for a foodie with lots of high end restaurants, but there are good and less expensive options around. Flights and rental cars at least tend to be relatively inexpensive.
Some dive destinations are simply cheaper than others to stay/eat/dive. Places where you can do some or a lot of shore diving can be less expensive - you (or especially your wife) might not be comfortable with independent shore diving to start, but house dives in front of resorts tend to be very accessible and easy and you might be up for some after you get sorted out. And if you're diving with an op on the water, doing training dives right off the dock instead of a boat can be a nice option and maybe less expensive.
Some of the cheaper places are more expensive to get to, so you always have to look at the bottom line from your location.
Bonaire is popular because of the shore diving, and even boat diving there is pretty cheap. Curacao is another place with lots of shore diving to pair with the boat diving. There's somewhat fewer convenient dive resorts on the water, but lots more options for everything, and the shore diving around the island might be considered more user-friendly. I haven't been to Cozumel but it's popular as a budget dive destination with many (even though not much for shore diving.) Roatan/Utila can be pretty inexpensive. There's lots of discussion of this sort of thing on this board.
Cayman does have some nice shore diving, and some resorts with good shore diving right on site - I don't think it makes sense to skip boat diving there, though apparently some people do. But it can reduce your average $/dive, anyway.