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We had runners on 1st and 2nd with 1 out. The batter hit a fly ball to shallow left center field. The CF came in and should have been able to make the catch, but dropped the ball (Didn't appear to be intentional). Because the runners had to stay on their bases, the other team ends up with an 8-5-4 double play.

Since the CF did not appear to intentionally drop the ball, I scored it as follows:

#11 reaches base due to an error on the center fielder #65.
#34 is forced out at third.
DOUBLE PLAY!

Is this correct or should the error not apply since the play ended the inning?

No error since an advancing runner was put out. The second out is a bonus.

I believe Batter is awarded a FC in this case since an out was made on another runner during his at-bat. I agree, no error.

It's not a fielder's choice at all. Fielder's choice would mean the Center Fielder could have thrown him out at first, but decided to throw it to 3rd.

It's either an error or a hit. I've never heard of it not being an error because an advancing runner is put out. That seems odd. If the batter got on because of an error, then so be it.

Depending on how far and how much effort the center fielder ran, I'd give the center fielder an error.

Doesn't error and fielder's choice do pretty much the same thing to the batting average anyway?

It is an FC. If a runner is forced out, then the batter is not credited with a hit and an error is not charged. That is the scoring rule and is not a judgement call.

the fielder's choice rule does not have any consideration on whether the batter would have been out, but whether an advancing runner was out on the play. Not always fair to the batter (i.e. a line drive to LF would most certainly be a hit with no runners on base, but with runners, a force out at third would result in a fielder's choice.)

NCAA Rule 14.21 - Errors (fastpitch rules)
14.21.3 An error shall be charged against any fielder when she catches a thrown ball or fields a ground ball in time to put out any runner on a force play and fails to tag the base or the runner, including a batter-runner on a play at first base.
NCAA Rule 14.7 - Fielder's Choice
14.7.1.1 When a ground ball is put in play and any preceding runner is out on the hit or would have been out had no error occurred.

agree...FC.