The sooner you pay off student loans the better, as they are a horrible burden. But do not deplete savings! 3-6 months of expenses is the MINIMUM for an emergency fund.
You'll need a substantial down payment for a car, too. If you don't put down at least 25% of purchase price, you'll have higher payments, higher interest rate, longer loan - all of which are trouble. Worst, a new car depreciates 25% the minute you drive it off the lot so you will be upside down in your car loan - owe more than what the car is worth.
Cut your plans on that car. Buy used, not new! NO lease as you have to get the most expensive type of auto insurance (double or more what you could pay otherwise). Choose an economical small car with excellent reliability ratings!
You remain heavily in debt despite your excellent work in paying off those student loans, but your emergency fund is a bit thin, especially if you plan to buy a car within the coming 12 months.