Have you done any research on Turmeric?
Are your periods regular? Are you overweight? Alot of women with the hair issue have PCOS.
I have some excess hair. Nothing major, but still a pain in the patootie. A few strays on my chin and upper lip. The ones on my chin are coarse hard hairs, while the upper lip are just darkened "peach fuzz". I started trying to pluck and then realized that, I take a super biotin every day and my hair grows VERY fast so I just couldn't keep up. I'd be mortified if I got to work and saw a stray hanging from my delicate feminine chin. I first started bleaching the hairs. Then one day I decided to try Nair. I have a Shower Nair and it fortunately doesn't cause any bad reaction with my skin. Leave it on for about 10 before you get in the shower, then wash it off in the shower and it works well for me.
There is a new cream out there that I believe is prescription only (believe it starts with a v...though I know that's not very helpful) that is supposed to be great for hair removal. It sounds like your doctor just isn't really taking the time to understand your concerns. What would've been the harm in her testing your hormone levels to rule out something like PCOS or other imbalance? And the average person can't afford laser hair removal.
Other issues can cause excess hair growth: thyroid dysfunction, adrenal hyperplasia. It could certainly be an excess androgen disorder, but I'd want to rule out the other things before I started researching natural treatments. How do you know how to fix something if you don't know what you're fixing? Saw Palmetto is normally used by men with prostate issues. There is so evidence that it helps with enlarged prostates. Anything else it claims to do, there is really not enough evidence.
I understand your frustration. :\ My recommendation is as follows:
1. Insist upon a hormone test which should include your thyroid.
2. Develop your plan of action based on the results.