Do you have a "protection" filter on the lens? If so, remove it and try without any filter. Might be better.
If it's the 150-
500mm APO OS HSM... are you certain the OS (optical stabilization) is working? Trying to hand hold without it will certainly have an effect on the sharpness of a lot of images. 1/800 is actually just about the bare minimum shutter speed to try to hand hold steady shots with an unstabilized 500mm lens on an APS-C camera. Even with OS, you'd likely still see some percentage of your shots spoiled by camera shake, trying to handhold such long focal lengths. Get a monopod at least... a quality tripod with a gimbal mount would be even better.
You mention
"I have been told I may have a "bad copy" of the lens, although I find that utterly ridiculous as I paid about $700 for it used." I doubt it's a "bad copy", though that's a rather high price for the 150-500mm OS lens used, so I'd expect it to be in top condition and fully working. Most of them go for around $500-$600 now (Adorama and KEH each have them priced in that range, in a couple different mounts.)
In it's day, the 150-500mm was a bargain telephoto. In fact, it and similar 120-400mm and bigger 50-500mm that Sigma offered were the first of the "bargain" super telephotos. Before Sigma produced them, you had to sell your car to buy a big telephoto!
However, it never was the sharpest lens. The past five or six years there have been a lot of new and improved, powerful tele-zooms introduced by both third party and OEM manufacturers since then.
Micro adjust might help a little, is worth a try. But switching to using the lens on a full frame camera you'll be giving up a lot of the "reach" you've enjoyed using it on a DX camera. For a birder, I'd have recommended sticking with a DX camera, unless you have the really big $$$ for the big, heavy super telephotos and a sturdy tripod to put them on.
The 150-500mm was discontinued and replaced by the Sigma 150-
600mm "Contemporary" in 2014. The newer lens has noticeably better image quality at all the focal lengths they share, is still in production and now sells for just under $1000 new. There's also a more "robust" Sport version of Sigma's 150-600mm that's closer to $2000, better built and sealed, but bigger, heavier, and not really much different in image quality. Tamron also offers a 150-600mm.... their current G2 improved upon image quality, build and performance of their first version, and sells new for about $1500.
In comparison, a Canon or Nikon 500mm f/4 lens will set you back close to $10,000 and are about twice the size and weight. One of the more affordable is the Nikon 200-500mm f/5.6 which costs about $1500... The the higher performance and more premium build/better sealed Nikon 80-400mm and Canon 100-400mm each cost around $2000.
You can compare image quality of the 150-500mm versus the newer 150-600mm (or other lenses) here:
https://www.the-digital-picture.com/Reviews/ISO-12233-Sample-Crops.aspx?Lens=683&Camera=453&Sample=0&FLI=3&API=0&LensComp=990&CameraComp=453&SampleComp=0&FLIComp=4&APIComp=0Note: the test shots at the above link were not done using the same camera as you.... But for comparison's sake, it doesn't matter so long as you set the site to display tests from both that were made with the same camera... in this case, a Canon 1Ds Mark III. That happens to be a full frame camera, like your D610, so gives you some idea what to expect on the FX camera. But when using the lens on a DX camera like your D3300, the corners and edges of those sample/test shots would be cropped away, the DX camera will only use the central portion of a lens, which is better than the corners/edges in most cases.