Factual error: After leaving Paul's apartment the first time Connie is shown running through Grand Central. She runs into a track tunnel, but the train's destination on the schedule board is displayed in blue, which indicates that the train is a Harlem line train (these headers are color-coded). However in the remainder of the film both Connie and Edward are shown getting off at a station next to the Hudson river which is served by the (of course) Hudson Line, which are displayed with green colored destinations on its schedule board. Long story short she's getting on the wrong train.
Factual error: Diane Lane makes a phone call from one of the stations in New York and only inserts a quarter. This film was shot after the charge for a public phone call was changed to 35 cents.
Factual error: The phone number Constance gives Paul has a 914 area code which is the Westchester County area code, a suburb just outside of NYC. In a later scene, we see Constance shopping at a supermarket presumably in her neighborhood as she is in the suburbs, but when she hastily decides to drive into NYC, she drives over an unknown bridge that would never take you from Westchester County to NYC.
Answer: Although Connie wanted them to go to Mexico and assume new identities, it is implied that Edward, who initially goes along with that idea, will likely turn himself into the police. However, there's no definitive ending given, so the audience is left to interpret the outcome.
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An alternative ending was shot in which Edward says goodbye to Connie in the car and walks into the police station. The director discusses this on the commentary track. The alternate ending is on the DVD.