Flight Instructor with student reported they descended to avoid traffic. The other traffic also descended resulting in a NMAC.

Narrative

Student pilot and Instructor pilot were flying the ZZZ VFR transition northbound with the intention of practicing the RNAV XXL approach into ZZZ1. We requested this with the south frequency and they seem seemed to accept the request. When we were handed off to the north frequency Approach ended radar services and gave us a frequency change as soon as they were able to. Assuming there was some confusion somewhere we continued to the IAF while I tried to reestablish communication with Approach and seek clarification. The frequency was so congested that I was not able to contact them again. Eventually we heard another aircraft request the same approach that we wanted to do; Approach denied their request because it was too busy. After hearing this we decided to make left turn and head back to ZZZ1. Once we completed the left turn we noticed a cirrus at our 9 o'clock higher heading our way. We decided the safest way to avoid the traffic was to descend. As we descended the cirrus aircraft started to descend as well. This pinched us between the ZZZ2 Delta and the traffic. We did not enter ZZZ2 airspace; but we got close. The cirrus aircraft did not make any effort to avoid us and continued along its way to ZZZ1 as well. Attempting to see and avoid traffic as well as avoid airspace put us in an uncomfortable position. I did the best I could and will try to avoid situations like this in the future.