Sounds like you are horsing the fish if you are snapping 300lb line. I caught a 40" sturgeon last year on 10lb braid Med Heavy walleye rod. Just set your drag real loose. When it runs, let it run. Only reel in when the fish lets up slack.
These are large powerful creatures. On a rod, they feel exactly like a snag by slowly spooling out line. The trick is to have tons of line and a really good drag system that does not bind up. If your drag locks up for even a brief second, you are toast. If you are using braided line like PowerPro, make sure you have at least 10ft of mono as a leader so the line will give a little.
Like I said, patience. It took me over an hour to bring that sturgeon in but hey .. I was targetting walleye with that gear. Thank goodness for double trolling batteries so I could follow the beast. Michigan is fertile grounds for Sturgeons.