I have been flying into and out of DCA for 20+ years, both corporate and airline. I was domiciled there for 4 years. It is no exaggeration to say I have gone into and out of there hundreds of times.
You stay on the tower frequency until instructed by tower to contact departure.
While acknowledging this very serious communication failure, I have found Potomac Approach to be competent and professional. I have found the local controllers at DCA to be some of the best in the business.
Without attempting to downplay the seriousness of the communication breakdown and potentially disastrous consequences that could have resulted therefrom, once the situation was recognized it was dealt with quickly and effectively. There was the potential for a collision due to a communication breakdown between TRACON and Local; there was not an imminent collision due to individual controller error.