The other expert has opted out.
There are several issues here, some of which can be interrelated.
The most recent ultrasound raises the possibility that adenomyosis is present. Adenomyosis is a condition in which the lining of the uterus grows down into the muscle layer of the uterus, and this can most often cause pain or abnormal bleeding. You do not note that your doctor discussed this with you, although you did mention that no fibroids were identified. If ultrasound is not able to make a clear diagnosis whether adenomyosis is present, then an MRI would be an option to try to obtain clear images of the uterus.
If adenomyosis is present, then the ultimate treatment would be a hysterectomy, which would resolve the symptoms associated with the adenomyosis and eliminate the need for discussion of contraceptives. If trying to avoid a hysterectomy, it can be tried to use hormonal interventions to try to control the adenomyosis. One of the newer treatments for adenomyosis that also may help to control the menstrual bleeding would be the hormone impregnated IUD.
As for depression, there are many people that notice that their depression is associated with hormone levels or hormone fluctuations. Many different patterns can be seen, so we will address whichever pattern is manifested in an individual. There is no direct evidence that Effexor is more prone to the effect of hormones than other antidepressants, but if that is what you have experienced, then we would still change to a different treatment.
It would be reasonable to only change one component of treatment at a time. If changing contraceptive and antidepressants at the same time, and there is a worsening of symptoms, it would be difficult to know which change was the underlying cause.
At this point, the usual approach would be interventions to address the possible adenomyosis, and that may require consultation with a Gynecologist, rather than further care from your GP.