What can the ohms of a DC motor tell me about itself

You should not use that driver if you think you'll be pulling that much current. For one, it's rated for 3A for a maximum of 10ms per the datasheet and you'll certainly try to draw for longer than 10ms in a stalled condition. And that is the maximum rating for the device which the datasheet says is not to be exceeded under any circumstance. It sounds to me like you'll be abusing the poor thing. Find a beefier driver, or if you're feeling really adventurous, parallel two of them. Is that a good idea? I've never done it, never heard of it being done, and wouldn't do it myself, but you could certainly try :)

I would just find another motor driver on pololu's site. It looks like there's plenty of them. Make sure their maximum CONTINUOUS current is greater than your stall current.

What do you use the push button for? I doubt you need such high current and can easily do with much less, especially when you use it as input for a logic circuit. Although @rawbrawb's 360 ohm will probably work perfectly well, it does draw a lot of current from the power supply. When using a 9V battery, the cell will be drained very quickly.

If the push button indeed is only used for a logic level to drive an opamp or digital chip, a resistor of 10kΩ would already suffice. You didn't share the circuit diagram, that might help in deciding on a good value.