Hi Lydia,
I’m no expert on Pusley specifically but there are folks in Florida who are! I imagine you have already tried what Anderson et al. (2021) summarized in the EDIS document on Biology and management of Pusley in Turfgrass and Landscape Planting Beds. I’d encourage you to reach out to one of the authors directly for any updates on management techniques. Perhaps Chris Marble (marblesc@ufl.edu).
The only other idea I had was to use a smother crop in the off-season when you aren’t growing crops. Mucuna and lablab are two potential ideas but be careful to not let mucuna go to seed as it has weedy tendencies. You can see how we used it as a smother crop to control weeds here: Can you plant a cover crop into grass?