Go get all of them that you want. They are there.
I used to snag a lot below the dam in the 90s, but once the construction started I couldn't fish the Marshall Co side like I used to. Once the construction was complete, the asian carp were getting so bad that it wasn't worth the effort anymore as you'd be worn out on carp before you ever snagged the first paddlefish.
Make note of the rules, as I often see guys in areas where they shouldn't be:
The area from Kentucky Dam to the new U.S. 62 bridge will be open to snagging 24 hours per day from January 1 through May 31. From June 1 through December 31, the area is open to snagging from sunset to sunrise. The area of the Tennessee River from the new U.S. 62 bridge to the I-24 bridge is closed to snagging year round. The area from the I-24 bridge to the confluence with the Ohio River is open to snagging year round. There is now a daily creel limit of 8 fish in aggregate and shall not exceed the daily creel limit for any sport fish in which the creel limit is under 8 fish per day. Snagging must cease if any sport fish daily creel limit is attained. All fish snagged including paddlefish, except shad, herring or Asian carp, must be taken into possession and not culled. Snagged fish must not be disposed on the bank. This action is littering and subject to a fine.