찡하다, 짠하다, 찐하다
어떤 슬픈 상황이나 사람을 보았을 떄 일어나는 감정, 또는 마음이 감동을 받은 모습을 나타내는 형용사.
When they describe the way something sad or moving creates an emotional reaction in your mind, I believe there is no notable difference. It's more of a personal preference which one to use.
찡하다 can also describe a dull, cracking sound, and 찐하다 might mean 진하다 (thick, concentrated).
It seems 짠하다 is also used to describe a remorseful reaction after doing something wrong, more than the others.
All of them describe a quiet and long lasting reaction.
For a sudden, strong reaction of sadness, we have 울컥 and 왈칵 ((마음이 울컥하다, 눈물이 왈칵 쏟아지다).