Will battery strength affect winch relays?

Second year with the Super windy 2000. I was plowing just fine, then the winch started acting up. It would not lift the plow. I had to hold the switch several seconds before it activated.
I switched the 2 relays around and the issue switched from lifting to lowering. So I figured one of the relays was bad. I purchased 2 new ones.
The new ones did not help. I checked the hoist connections and all were good. I had a trickle charger on the battery all this time.
I put a larger charger on the year old battery with a 50 amp boost. The plow moved up and down but there was still some hesitation when pushing the switch.
I put a load tested on the battery. 13.4v before the load. The load pulled it down to 10.2, and then it read 12.8 after the load.
Does that.e battery need much juice to engage the relays?
Got any ideas for things I should check?

Pre load test….

Load test…….

After the load test…..