Perhaps your initial contact could say something along the lines of, "Hey I am interested in looking at your car but I only have $7,000 to spend. I don't want to waste your time, so let me know if you could let it go for that much and I would love to come check it out."
This sounds like a good strategy to me.
Recently I've been selling LOTS of stuff on CL, and as others have said above, you get some weirdos, but there are also many good people who are just looking for a better deal than they can get in a store.
A month or so ago I had a nice bed listed on CL for $850. At least 6 people contacted me during the first week I had it listed, and they all "definitely" wanted to buy the bed because they loved they way it looked in the photos. When I talked with the people on the phone they asked me questions like, "Is the bed comfy?" or "What kind of mattress is it?" But, none of them asked important practical questions like, "What are the dimensions of the bed?" (huge) or "How much does it weigh?" (heavy) or "Can the frame be disassembled?" (NO!) Which were the most important questions to ask if you were actually thinking of the logistics of buying and transportiung this giant, solid, massive bed made entirely of super heavy tropical hardwoods.
After a week of talking, texting and emailing with all of these really "nice" people who didn't have a clue, I got a text from a guy who said, "Hey, your bed looks great. I'd love to buy it, but our budget is only $500. If you'd be willing to consider accepting that much for your bed, please let me know." We texted back and forth a few times. I told him if he actually showed up at my house with a truck ready to load the bed up and take it away, I'd gladly accept his $500. This guy asked all the right questions. He wanted to know the exact dimensions, so that he could measure his home doorways and make sure the bed would fit into the bedroom where he planned to put it. Less than 24 hours later the guy showed up at my house with $500 cash in hand. I helped him load the bed onto his truck and he drove it off. I was more than happy to accept a lot less for the bed than I'd initially asked, just because the guy showed up and had the cash.
I don't agree with the advice above to, "Not make low ball offers on CL." When I put something up for sale on CL, I want to sell it. If I think a fair price for something is $100, I'll usually ask around $120. Sometimes people just pay the $120, and I end up with a little more than I thought I'd get. Usually people will want to negotiate a little bit, so I'm fine to drop the price a little bit down to $100, and then I end up with the amount that I thought it was worth to begin with. But, sometimes I put something on CL, and I think it's worth $100, but I don't get any offers, so I drop the price, drop the price again, and again. In the end, I've taken $10 for stuff that I originally asked $100 for, but I don't care. I'm not offended. The price of anything you sell on CL is whatever the market will bear. If no one's interested in whatever you have for sale, then it's worth $0. I don't care what Kelly Blue Book or anyone else says.