You'll get 1/80s with the same blur since you have the same field of view. Depth of field will be different though, the FF will be shallower.
If you stop down the FF to f2, then you'll get similar depth of field, but then you'll have to raise ISO to compensate.
If you're seeing motion blur with a speedlight, it's because your ambient light is too much. The speedlights flash duration will be much quicker than 1/160th and freeze motion on it's own. You need to let the speedlight expose your subject, not the ambient light. If that's done properly, you'll get motion freezing when shooting lower shutter speeds like 1/80th or even lower.
I almost always do manual flash power, and always always when using them off-camera so I can really dial in my exposure for the flashed subject and ambient light. I only use ETTL when shooting in lighting conditions that change frequently.
High speed sync on your speedlight should allow you to shoot faster than 1/160th provided your lights are properly communicating with your camera, but sounds like that's not the case, and it does affect power output, so you should turn it off since you aren't making use of it.