If I am still having regular menstrual cycles while on tamoxifen, is there a chance the tamoxifen isn't really doing it's job?

Normal to cycle. Tamoxifen is an estrogen receptor modulator and is not intended to suppress the ovaries. It is intended to block the estrogen receptors so that estrogen cannot bind and stimulate that particular tissue - usually tamoxifen is used to block stimulation of breast tissue in patients with estrogen receptor positive breast cancer. Tamoxifen is not intended to stop you from having normal cycles.
Possible. I would talk to your oncologist and gynecologist about this. It depends on your age and weight, etc. There are blood tests that can be done to determine how well your ovaries are being suppressed.