Since we had no plans to look for a new car, we do not have adequate savings to purchase another used car, as much as I would like to.
Do you have an emergency fund? The time to save for your next car is before you need it, of course.
A friend of ours recently had the same problems. Her car was going to cost too much to fix and would likely not be running well in only a few short months. She got her car appraised for a trade-in value and as given more money back than she expected. She decided to lease a new car, using the trade-in money as a down payment and with a low monthly lease.
Your friend may or may not have gotten a good deal. When you trade in a car for a new one, be it purchase with loan or lease, they have a lot numbers to play with. Look up the "four square" technique, e.g. http://consumerist.com/2007/03/30/dealerships-rip-you-off-with-the-four-square-heres-how-to-beat-it/
So for example, if they sense you want a lot for your trade-in, they'll give you more than it's worth, but then will increase the price of the car you're buying, or the interest rate (leases do have an interest rate). Or in the case of a lease, they might be able to lower the residual value.
Leases are generally not a very good deal. In general, the average lease is going to cost the consumer more than the average new car purchase, and the average new car will cost the consumer more than the average used car purchase. There are exceptions, but they are exceptions to the general rule.