If you're trying to move significant weight in the weight room, you need to kick off your running shoes before a big set. As an athlete, you're trying to put as much force into the ground as you can. That's how you move the weight, whether it's a deadlift, a squat, or a lunge. Running shoes slow that down.
Why? Your running shoes are designed to absorb energy. Usually, it's the energy from the road going into your feet when you run. In the weight room, you're on the offensive! You're the one creating energy that's driving down into the ground. The same way the shoes were helping on the run, they're now hurting you in the squat rack.
Are you looking for a cheap and easy solution? Just go barefoot. Leave your socks on and lift without your shoes on. There won't be anything to stand between the force you're creating and the floor. I've had hundreds of high school athletes set a PR their first day in my gym, just by telling them to go barefoot on their big strength moves.
Are you looking for a pair of shoes to wear in the weight room? Look for something with a hard rubber sole. Old school Chuck Taylor's work great. Wrestling shoes work great. Slide those on and get to work!