I had this same problem a few weeks ago, and it is probably the starter.
I had to have mine replaced with a new one. Same symptoms too, battery and alternator checked out fine, when I finall got it started, but the starter was totally going.
Also, had some existing problems even after I got the starter, and I had to replace the battery cables, they looked O.K., but they were corroded and worn.
You may want to check that to.

Hope this helps

Here's what I do when bitten by the no-crank bug.

1) With the door open and thus the dome light on - hit the starter. Any change in light? If it goes way dim, the starter is getting power, but not enough. If the light doesn't change, the starter isn't drawing any power.

2) There is an insulated female connector under the hood, usually near the firewall, that is used by a tech to engage the starter without using the key. If you can get a grounded wire into that connector and the starter remains inert, then the key and shifter switch are innocent.

3) Look over the wires connected to the starter. A big fat one comes direct to the battery and it's usually OK. A thin one is the one carrying current to engage the solenoid and thus the starter. If it is bad, you get no-crank.

Good Luck,

Bob

:>)

Id be checking the starter right off the bat, take a hammer and hit the starter with it, if its stuck you should be able to unstick it that way a few times. If it starts after you wack it a few times you know the starter is going out.

Good Luck