Ooooh, if that's the case, Card Sound Road may be a better choice.
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She won't get it on time online, unfortunately. And the stick-on one can't be moved from car to car. And while Card Sound is longer miles-wise, if the 18-mile stretch is under construction, it will take longer to drive it.
And the FL City>K.L. stretch should be OK on an 11:10AM weekday. DebbyDiver's 5-1-1 (thanks for that DD) would be a good tool to use just to be sure.
I'd also add there's an "amber alert" sign at the fork between CSR and the Stretch. When there's a traffic issue, that sign will direct you which road to take.
NO NO NO do NOT do the stop and pay, you will kick yourself. It's going to cost you at least 20 minutes each way of your life that you'll never get back. Get the Sunpass.
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It's worth it even if it costs you a few bucks. Let's face it, do you want to spend 20 minutes or more waiting in line to pay a toll (approx. 5 minutes per tollbooth and there are four), or spend that 20 minutes in the Keys? Me wuvs Sunpass!
Sunpass website
595 is the easiest and has the most signs directing you to the turnpike. I'm thinking a 9:20 FLL arrival translates to a 10:15+/- departure from the car rental center, putting you after most morning commuter traffic. And the FL City>K.L. stretch should be OK on an 11:10AM weekday.