Don't think travel will make everything better. Don't think it'll never get old, either. I don't miss food from home any more or even craft beer, which was big at first, and I've had a GREAT experience here. I LOVE my students. I'm saving actually nearly as much as I would be at home, and I'm living a higher lifestyle here. But it's not the end-all be-all. Being away from family/friends gets old, making new friends who cycle through the city gets old, missing hobbies you just can't do here gets old, etc.
This. I've been teaching in China for nearly 2 years now. I'm just now ready to pay off my final student loan solely from the money I've been able to save at this job. I was dead set on staying another year and simply saving (I can save ~$1,500-$2,000 a month) so I would be totally set when I decided to leave so I could travel, maybe bum around Europe.
But now... I'm incredibly homesick. The last two months have been rough. The short-timers cycled through and the few long-term expats I connected with are moving to other cities or leaving. I miss home. I miss my friends. I miss not everything in life being so difficult. I miss feeling like I'm accomplishing something and moving forward. So here I am, 90% set on going home at the end of my current contract (mid-June). I left the US not planning to ever come back. Things change.
I know someone who moved here with $1k in his pocket. It was stressful, but he made it work. I was comfortable moving here because I had $20k in savings. The point is: you can support yourself pretty easily teaching 10ish hours per week at $20/hr (rates in other countries might be different). What you feel comfortable leaving your home country with is up to you.
Also this. Definitely this. I had a good cushion saved up and was mortified when one of my coworkers showed up to China with only $600 to her name. If I had realized how easy it was to get teaching jobs as a native speaker, especially in Asia, I wouldn't have panicked nearly so much. The cost of living is low. The demand for English teachers is high.
Basically, a lot of your budget depends on your style of travel. If you're doing a lot of country hopping, lots of flights and short stays, it will be more expensive. If you're going slower, taking buses and trains, staying for the duration of tourist visas... You can take advantage of a lot of lower costs (i.e. sublet an apartment for a month, rather than rent a room) and can always find a way to earn some extra cash if you find someplace you want to stay longer.
For a year, I wouldn't plan on less than $10,000. My personal rule when I left was to always to have enough cash savings untouched that I could buy an emergency last-minute flight home. Your comfort zone may be bigger or smaller than that. I also have a couple of high-limit credit cards I could put a flight on if the situation were absolutely dire which helps with ease of mind, but my cash has never gotten low enough that I've even needed to consider that. And you can't use credit cards most places in China anyway.
Also if you need any dental work done, do it in Thailand. ;)
ETA:Forgot to address the student loan/car thing. My student loans are on a graduated repayment plan, though obviously I've been throwing significant extra payments at them. The logic was to have the lowest possible necessary payment at any time so if something happened and I didn't have the cash to cover a larger payment, I wouldn't be screwed and trash my payment history/credit. I never feel comfortable unless I have about 6 months coverage of ALL my US bills in my US checking account. This includes absolutely everything that gets regularly billed to my US accounts. Part of my reasoning is that if something were to go wrong, it would be fairly difficult for me to send emergency cash home quickly.
I would definitely sell the car, but I don't understand the logic in buying a new (used) one if you're about to leave the country. If it's at all possible, I would just try to go carless before you leave. Bank the savings and put it towards your trip.
Also some money saving tips: If you don't already have them, set up an AirBnB and Couchsurfing account. Both are widely used worldwide - AirBnB is even catching on here in the boonies of China. Both of these tools will save you a lot of money traveling and are a good way to meet locals or expats who can show you a much more interesting time than the general youth-hostel-hopping scene. Unfortunately, it can be hard to get your first couple of stays if you don't have reviews to show you're a real person/not an axe-murderer, etc, but it gets much easier once you've been reviewed a few times on the sites. Ask around if any of your friends are members, and at least for Couchsurfing they can confirm knowing you and write you a recommendation. AirBnB I think you actually have to stay with someone for them to write you a review; i.e. a payment has to be processed.