@whahoo, From what I understand the level can naturally rise and fall. Some oncologists feel that rising results can be alarming for patients when there is no new cancer. Others believe that the tests can be an early sign that something may be headed in the wrong direction.
I do not have CEA tests. My oncologist generally doesn't order them for his patients. If offered, I would probably have the test done. I do have continuing scans (every 3-4 months), and I don't tend to react to every little comment in the reports, so I think I could handle the rise and fall in numbers.
Are you getting CEA tests?
https://my.clevelandclinic.org/health/diagnostics/22744-cea-test-carcinoembryonic-antigen