Pilot of a single-engine aircraft reported taking evasive action following a TCAS warning in order to avoid traffic visually acquired following an ATC alert.

Narrative

I was descending VFR towards FTW to my assigned altitude of 3;000 ft. Regional approach advised me of Aircraft Y that just departed FTW and leveling at 3;000 ft. As I was passing through 3;500 ft. the Controller asked if I had traffic in sight at which time I did not visually; only on my TCAS. He told me to stop my descent at 3;500 ft. but I was only able to catch my level off at 3;400 ft. due to the having already passed through 3;500 ft. I then picked up Aircraft Y turning right; on a convergent path with my aircraft. I told ATC that they were in sight and it looked like a convergence problem and I told him I was descending. Aircraft Y was turning right in my course and after I made a rapid descent to 2;500 ft. MSL I saw the traffic pass approximately 400-500 ft. above me in a westerly direction as I kept the same heading towards the FTW airfield. Not once did I hear Aircraft Y say he had me in sight or respond to my convergence comment to ATC. I responded to a visual traffic avoidance and an RA in my instruments. There was light traffic activity in the area at the time so I do not believe that task saturation played a part for any party involved. If Aircraft Y had not seen me visually they should have relied on their TCAS for situational awareness as they were turning into the downwind environment from their departure. I do believe there should have been some acknowledgment on Aircraft Y's behalf after I made the convergence call.