Month to Month Tenants Not Paying Rent Increase

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I recently purchased my first investment property (a duplex where the rent hasn't changed in 6 years with two month to month renters). I intended to do work and renovate building + occupy one unit. I gave the proper notice to terminate their leases and the tenants said they needed more time (but would not agree to a date to move by). Since this is delaying my ability to make improvements + I need to pay extra for my lease to go month to month I provided a notice of increase in rent. One tenant says they are not going to pay the new rent they are going to continue to pay the old rent - they specifically said they want to settle this in court. Evictions are on pause in St. Louis through the end of the year so I can't begin the eviction process. Any advice for how to proceed? I'm looking into lawyers now but want to see if anyone has experienced this.

My real estate agent suggested cash for keys but I am worried this will confuse the tenants and they will try and "use this against me" in court. 

Thank you for your help!

Rachel