DA20-C1 Pilot reported aircraft shaking during flap retraction resulted in a safe return for landing. Pilot also reported lack of attention to pattern traffic resulted in an airborne conflict.

Narrative

Day 0; I went out on a flight with a friend to practice my instructing skills as I am currently a CFI student. We went southeast of the airfield; and I practiced teaching various maneuvers. Then we came back to ZZZ; our origin airfield; for some pattern work. After my first touch-and-go on Runway XX; at 500 ft AGL on the upwind; I raised my flaps to cruise configuration; and the airframe began to shake violently. I visually inspected both flaps; and they appeared to be normal. In response; I returned my flaps to the T/O configuration they were previously in; something that isn't typical procedure in the DA20-C1. I made the decision that I had to make my next landing full stop; as this was an unfamiliar situation and not something I had ever been trained on. While in the downwind; I tried troubleshooting by experimenting with various flap configurations; but the cruise flaps setting was still consistently producing violent shaking. This first hole in the Swiss Cheese Model; admittedly; led to considerable task saturation. I became fixated on gaining altitude and airspeed while still having flaps in due to the fear of causing the shaking to occur again when I raised my flaps; meanwhile ensuring that the engine was functioning normally. In this highly task-saturated downwind leg; I failed to take note of an arriving Citation CJ3+. They entered an extended left base for XX; and as they began to establish on short final; I began to enter my left base for XX. Upon turning base; I made visual contact with the Citation and pitched up to avoid getting in its flight path. If I had continued on my descent path without seeing them; it wouldn't have resulted in a mid-air collision; however; it would've been too close for comfort. The approximate closest distance between us was about 1500 feet horizontal; 500 feet vertical. This is a great lesson on how high-stress situations; such as abnormal shaking of the airframe due to a flap issue; can result in the inability to make accurate perceptions. As an instrument-rated commercial pilot; this mistake is an embarrassing but incredibly valuable lesson. The impact on perceptions that the stress of an abnormal situation in flight can have on the pilot is drastic. This was a wake-up call that taught me the importance of maintaining traffic separation at the top of my priorities- aviate (avoid traffic); navigate; communicate. Ultimately; this mistake served as an incredibly valuable lesson in showing me how I personally respond to alarming situations in the air. I'm grateful that everyone is safe and that I can move on from this situation as a better pilot. I look forward to ensuring my future students don't make the same mistake.