Do you change your career or careers?
Solution 1:
You say "I changed shirts" or "I changed my shirt".
Similarly, "I changed careers" or "I changed my career".
Of course, the "my" construction can be plural:
I was wearing two shirts, then I changed my shirts
I was a lawyer and a programmer, then I became a bus driver --> I changed my careers.
You would not say "I changed shirt", so "I changed career" is also not grammatically correct.
Having said that, people do say it, but this Ngram shows you that "changed careers" is by far the more common way.