And keep mentioning how many people are relying on this deadline in conversations - You, your new tenant, their current landlord, the new tenant there, your handyman, locksmith, cleaning crew, etc.

I actually wondered myself if this was the same guy mentioned by @Robert Abasolo in one his videos and in [ Tiny Houses, Interior Design, & Drunk Locksmiths w/ Robuilt - YouTube ] because he acted the same exact way, and used the same expression ("I got you . . .

Can I have a locksmith drill out the locks and enter the property like the bank does in order to winterize and protect the property? 

You also need appropriate language in your lease or house rules to authorize (and reference) these fees.Examples could include- auger toilet; rekey locks (per cylinder); painting (per room, or SF): cleaning appliance; stripping/waxing per room; carpet shampoo per room; light cleaning; heavy cleaning; trash removal per pickup load...You also need an adequate disclaimer for NON-routine situations- perhaps an hourly charge, or emergency after hours rates, etc.

Locksmiths, 4. 

This enables you to rekey a unit every time in just a couple minutes.

- Product Dimensions : 4 x 3.8 x 6.8 inches- Integrate the lock with a z-wave hub and then enable that z-wave hub skill from the Alexa appKwikset Smart LockFeatures:- Keyless entry touchpad with a motorized deadbolt and sleek interior; includes traditional key functionality for peace of mind- 10 digit backlit audible keypad, one-touch locking, 30 second autolock option, and 16 customizable access codes for increased security- BHMA Grade 2 certified, UL certified with 20 minute fire rating, and features SmartKey re-key technology with BumpGuard protection- Prior to purchase, please verify that this lock fits your door- Fits standard doors (1-3/8" - 2" thick).

. - and that a locksmith has been engaged to change the locks upon their departure. 

Tenants can pay me $50 and I'll bring them a spare key, or they can hire a locksmith for $65.

If it was me and our state laws if no one answered I would show up with lock smith show them my sherrifs recipet and change the locks and leave my card that usually gets a call back.but in some states you cant do anything until the sherrifs deed is actually recorded which can take 30 to 120 days depending on county.. you really need to bone up on the rules there and what you can and cannot do without getting in trouble..