Best running shoes for hybrid training when your workouts are never just running

I’m getting more into hybrid training and I’m realizing my running shoes aren’t really cutting it once I start adding strength work.

A normal run feels great, but then I’m doing sled pushes, squats, lunges, carries, box jumps, etc. and I start wishing I had something more stable. At the same time, I don’t want to wear a flat gym shoe when I have a 5-6 mile run in the same session.

What are people using for this kind of training? I’m looking for the best running shoes for hybrid training that can handle both without being terrible at one side.

I’m fine with some compromise. I just want something I can leave on for the whole workout instead of changing shoes halfway through.