Billy -
I actually use a dumbbell anchor. It is not on a dog leash (although I did have a drag anchor for my kayak on a dog leash) - just on 550 cord that I spool around a hand line reel (big yoyo reel). When I get to a spot that I think/know holds fish, I throw the anchor, clip in and start fishing. I don't do this in fast current where swamping could occur (like the bottle neck under the Chain Bridge on the Potomac), but slower currents are no issue at all (like that off Fletcher's). You can use a dog leash to manage the line more efficiently, but if the water is deep (like off Fletchers in some spots), you'll still drift. In either case - don't forget to have a sharp knife with you in case you get hung up and have to cut away. Another way to keep in place is ride up on to the weed beds (summer time). Or if the wind is going against the current, you can use a sea anchor/drogue. I use all three (anchor-drogue-weed bed) depending on the conditions.
My board is an aircraft carrier (google Hooked SUP, or life amphibious + hooked sup) and I can strap down a cooler (storage and a seat), I have two scotty mounts that hold multiple rods and two bungee grids that I tuck the odd stuff under. If I were in the market now, I'd lean towards the rotomolded SUPs - you can beat those up agains rocks and not have to worry about dings - just like kayaks. My board is tough but not that tough, but it is also much lighter. I had a more "all-around" do it all type board earlier last year that I sold in favor of the fishing specific board - go in this direction first and you will be much happier. I take mine out into the lower bay (Virginia Beach) during the warmer months and it is the perfect mothership for fishing (kids have their own boards and they come back to me for drinks, snacks, handing off their fish or crabs, or if they need help rigging - I spin and fly fish with complete stability).
There are a few guys here on the forum that have fishing SUPs - there was a get together last spring. There are a bunch more that have kayaks. If you have a specific question - I'm sure you'll get tons of answers (and everyone has an opinion - pick and choose what will work for you).