The toilet runs when we fill the bathtub. The float is fully lifted.

How does the toilet "run "?Surely not flushing.

I am going to assume that the ball cock is what runs when you turn the cold water onto the bathtub. This is caused by fluctuating pressure fluction and waterlogged air chambers.
The solution is to recharge the lavatory air chambers. Air chambers are there to cushion the shock of water pressure fluction. In my area we are mandated to install air chambers on all lavatories and kitchen sinks. Over time they build up with condensate and must be recharged.
To recharge your air chambers. Shut the water off to the house and go under the lavatory to the angle stops. Remove the bonnet packing nut located under the handle and turn the stamina washer assembly out counterclockwise. There will be some water draining from the supply and the air chambers. Do this on both angle stops. Reassemble and test. Does it work better now? Good luck, Tom