Light Transport
Flight Crew Air Transport Pilot (ATP); Flight Crew Multiengine; Flight Crew Instrument
Communication Breakdown
ATC Issue All Types; Conflict NMAC; Deviation - Altitude Overshoot
General None Reported / Taken
Human Factors; Procedure
Departing FXE on the FLL7 Departure had leveled at 2;000. Controller gave a climb to 3;000; announced traffic and handed us off to the next Approach Controller. At this time the pilot flying announced less than 100 feet vertical separation from the traffic and we received a TA annunciation.  Pilot flying increased the rate of climb and we exceeded the cleared altitude by 300 feet avoiding the traffic.Suggestions - Pride [Provide] proper separation and don't hand off an aircraft to another Controller when there is a possible traffic conflict.
Light Transport aircraft First Officer reported an NMAC during departure from FXE airport and cited Controller handoff procedures as a contributing factor.