My first rental. Tenants are not leaving after lease, please help

[Paul Tibok]

Hello BP, new landlord here.  I need advice.  My property is in Fl. I bought my property in August and took over the lease for the previous owner.  My tenants lease is up Jan 31.  They told me they were leaving last month which is fine with me for many reasons but I have a feeling they  wont be leaving.  Its clear they havn't started packing and I just have a feeling they can't find a place or mabye now realized even tho I raised their rent in the new lease proposal they wont be able to find anywhere that compares for the price. I have already started to market the place and have plenty of people who want a viewing as soon as it is vacant.  What should I do besides talk to them and try to work it out if they are not out by Thursday?  To my understanding I need to give them a notice in person which states they have 72 hours to vacate, and if they don't then I need to file an eviction at the courthouse.  Is this accurate?  Aren't they trespassing after the lease expires?  If they write me another rent check because they say they want another month should I not accept it?  I want them out because they were not willing to pay the increased rent which was still under market but anyway please help me this is my first rental property and I need to go about this professionally and legally.