If your cyst is large enough, it can be felt by a competant physician during a vaginal exam. They may have to do a rectal exam too because the ovaries are better felt through the rectum. But that is only if the cyst is large enough. And it will in no way tell them what kind of cyst you have.
Functional cysts are extremely common, but if you are having constant pain and it has been going on for months then you really need to get an ultrasound and see what is going on inside.
Not all cysts are mucous. I also had a dermoid removed a few years ago and it was full of all sorts of nasty things. I had mine removed six weeks after the initial u/s that found it and I still lost 90% of my ovary.
Currently I have two masses in my right ovary. They are causing such debilitating pain that I cannot get up off of my bed or couch most days - let alone enjoy my life or my kids or anything. They are solid and my gynecologist and I won't know for sure what they are until they are removed in another week or so. They are sizable enough that we are very concerned about a torsion. Because they are big and solid and have grown very very quickly, the word "cancer" has come up in many conversations with my physician. The chance is small, but without knowing some of the information we do from u/s, he might not be as aggressive with these masses as he is.
If you are TTC, you owe it to yourself to get an ultrasound and see if what you have is something that could hinder the process. The last thing you want is to ignore it further and risk a torsion or ruprture or other complication that could impair your fertility forever.