I had a similar issue when I was looking to get my 590A1. I checked all over the net and couldn't find them in stock anywhere. Checked my local shop and they had 2. A friend and I bought both of them.
It seems that the 590A1's come out in batches like once a year or something. If that's what you really want, keep looking and they'll pop up eventually. As a 590A1 owner, I would probably suggest going with the 590SP. It's basically the same without the super heavy barrel, and that heavy barrel does make a different in the shotgun.
I do love the gun though and I'm used to the weight. I've got a pretty nice 870 Tactical as well that I like and have upgraded to all of the 'Police' parts, but I'd take the 590A1 any day. The 590SP would be an acceptable replacement if you can find one of those.
To more directly answer your question... yes... the panic buying has had a sizeable impact on the shotgun market as well. Don't get excited... just browse the usual spots and you'll find something to suit your needs soon enough. I'm a fan of slickguns.com
Glock 19 G4 Gray | Glock 17 G4 FDE | S&W M&P 22 | S&W M&P 45 | S&W M&P 9 | Sig P226 MK25 | Sig P229 Extreme | Sig P227 Carry | Sig P320 F | Sig P320 C | HK USP 9 | HK USP 45 | HK USP 45 (Tan) | HK VP9
Daniel Defense DDM4 V4 | Arsenal SGL 21 | ASA SBR | Savage Mark II FV-SR
Mossberg 590A1 20" SP | Mossberg 500 Cruiser