So many of you were so great with giving me advice and sympathy about my bad dye job that I wanted to come back with an update.
I spent a week shampooing my hair with Dawn dishwashing soap and clarifying shampoo (I followed up with an excellent conditioner so that my hair wouldn't dry out). After going back to my stylist again to get his opinion on what else could be done, and not feeling confident with his response, I went to a woman who had good reviews on Yelp for fixing bad dye jobs.
I went to her yesterday and she put a product on my hair that she said was designed to remove the dye, but leave whatever natural color underneath was there. She wasn't sure how much dye it would remove, but we were lucky because it got all of the wine-colored "red" out of my hair. Unfortunately, although my original stylist had told me that what he put on my hair was demi-permanent and had no ammonia or peroxide, there must have been some ammonia in it because my hair underneath was orange. The good thing about this is that it was really light, so we had options in terms of getting me back to my natural color (the color the original stylist had given me was much darker than my natural color).
The colorist put a brown gloss on my hair, but didn't leave it on for the entire time, because she didn't want me to end up too dark again. I actually think that the color looks pretty good! It's a little lighter than my natural color, and has more copper and gold in it than my hair usually does, but I'm liking it! In keeping with our discussions on color on this forum, the key here is that the new color is very warm, so it matches my skin tone. The bad dye job was VERY cool in tone, and my skin looked really yellow and sickly next to it.
Thanks everyone for the support when I was so miserable!