C560 Captain reported trying to obtain IFR clearance in the air after being unable to while on the ground and shortly after departing there was an RA that the reporter responded to but resulted in the aircraft overshooting into Bravo airspace.

Narrative

I filed an IFR flight plan ZZZ to ZZZ1. I was unable to raise ZZZ Departure on the ground despite numerous attempts. No NOTAM was present regarding the frequency XXX.X being unusable; and this was the frequency listed on Foreflight which was current. I troubleshot the radios; connections and headphones; but still unable to contact ZZZ Departure to open my IFR flight plan. There was no phone number in my Foreflight info page to call except for FSS ZZZ. I decided to depart VFR and pick up IFR in the air. I departed Runway XX; conditions were clear VFR weather; with a fair amount of GA traffic. After takeoff; I called ZZZ Departure but no answer. Shortly after departing; I received a RA instructing me to climb; climb; climb. I increased power to full power and climbed and saw a Cessna 172 or 152 close enough that I could see the pilot. I estimated 500 feet away; 200 feet vertical separation. It was the closest I had been to a collision; and it was very unsettling causing an adrenaline response. I climbed to avoid a collision with full power. I overshot into the Bravo airspace and immediately attempted to contact ATC at the Departure frequency in Foreflight XXX.X. I tried both radios; confirmed switch on headphones was on and not to the oxygen mask; but still unable to contact. I went to 121.5 and asked for any pilot to provide ZZZ Departure Control frequency and was able to reach ZZZ Departure.This was a preventable situation; and if I had been more persistent in calling FSS I believe they would have been able to find a phone number for me to call ATC before departure. Given the busy GA traffic on a nice day; it would have been much safer to obtain IFR clearance on the ground prior to departure. The second factor is that after climbing out at full power in response to the RA; I should have focused not on calling ATC but on getting the aircraft out of Bravo airspace. This was a very preventable event which should not happen again if I follow these recommendations.