When is the best time to sign

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The buyers want a final walk through at 10AM Monday morning, then go from the house to the title company to sign. I have a padlock on a window there yet. I want to get it Monday morning. That shouldn't be a problem. I don't want to remove it before then. Or, should I wait until after they walkthrough and sign? I don't think there's any problem, I'd just feel better knowing the house was secure to that point. I planned to take all the keys with me to the title co and leave them there. Should I wait untill the transfer is recorded? Closing may not be til the next day.

Apparently I may have the opportunity to sign before they do--earlier on Monday morning, or later in the day, even Tuesday.
Is it better for the seller to sign after the buyer, or befoe? Does it really matter?
Thanks.

FWIW, I accepted an offer of 15K less than what I had hoped to get, because of what I saw as market conditions. I doubt that I could have gotten much more anytime soon. I used a flat fee lister, and the buyer agent had no problem with that.