FYI, my photo experience is mostly above water. That said, as much as I value having image stabilization, in general, it seems less important under water. OSS can dampen tiny hand movements, but it does nothing to help the blur you get from drifting/floating nor does it help freeze anything in your frame that's moving. In other words, most of the time you're going to have to increase shutter speed enough to stop motion, and at that point OSS does nothing for you at all. Stabilization is most useful for shooting static subjects handheld in low light. If you want your camera and lens to do that sort if photography for you in addition to underwater stuff, then prioritize OSS.
Sony has made a strategic decision to focus on full frame photography, at least for the foreseeable. The quality in their APS-C lineup is often lacking, and Sony hasn't even brought forth a new crop lens in years. The 16-50 kit lens isn't so great, which is a shame because it's the lowest common denominator for underwater housings.
Fair disclosure: I'm a pretty harsh critic when it comes to photography gear - a lot of people are perfectly happy with that kit lens, and it's probably the only lens most people ever even buy. But, when you're noticing that a lens isn't cutting it, as you remarked, you're chasing an above normal image quality, and the kit lens won't get you there. Part of the problem is that sensor technology has outpaced lens production technology -- especially with Sony because their sensors are stellar. Part of the problem is that Sony is only putting their best efforts into full frame. And, part of the problem seems to be Sony's own difficulty in making good glass with low production variances. Be wary of chasing lens quality, though. There is no bottom to the lens quality rabbit hole. I only have a couple of lenses that cost under US$1K, and a lot of people are way farther down the hole than I am. (My good gear stays on land.) I only mention that because what it takes for a lens to look good to me may be above and beyond the call for you. I've pretty much abandoned APS-C e-mount glass.
If your lens is 'really-OMG-bad' you might have a bad copy (or damaged). It's hard to test well unless you've tested a lot or have another copy to compare, side by side under controlled circumstances. I don't own it, but F8 should be about optimum on that 16-50 for across-the-frame sharpness.
Sony APS-C lens ideas (without getting stratospheric): Understand that individual copies of lenses aren't identical, like we consumers have come to expect in almost everything else. There's a lot of copy variation. Read up on how to test lenses and decide how far you want to take it. You can save money buying used, but know that there are more used lemons, too.
A lot of people like the 35/1.8. The copy I tried sucked. Badly. Try the 35 Sigma. No OSS but sharper. (Same for the Sigma 60, but I'd rather have the FE55 if it fits in the housing.) No personal experience with the 30M you mentioned, but I think I'd want something longer. Please post how it goes. The best macro is the 90/2.8, and a lot of 6300 housings take it, even though it's full frame. The Sigma 19 might be worth trying because you should skip Sony's 20 and 16 altogether. No personal experience with the 19, though. The Sony 24Z is terrific - one of the best APS-C lenses for Sony's or any other crop body platform. Priced like it. Not sure many housings accept it. For wide, I had a lot of fun with the 10-18, and many underwater housings take it. Primes almost always make better images than zooms, but still this is a great lens for the APS-C platform. (Try to get it to f8 (5.6 will do) and avoid the long end.) The Zeiss Touit line is good, overall; just overpriced. Probably lower copy variance. The Sony 16-70 is a little better than the kit lens you have, but there's a huge copy variance problem, and even then it's fairly worth half the price they charge new. I can't recommend it as a solution, really, and few housings take it, anyway.
This has grown long, so I'll stop there, for now.
