How to recover files from a crashed hard drive

I believe my hard drive is in the early stages of crashing. Last night I was using my Macbook Pro and was running safari and microsoft word when all the sudden the computer froze up and I got the spinning wheel. I waited for about 5 minutes hoping the wheel would go away and after no avail decided to turn the computer off by holding the power button. I waited a couple minutes to let it cool down and then I rebooted it and I was then given the jumping folder with a ? on it. I did some research and found that this basically means your hard drive was crashing. So I read a solution online that if you hit the Option button while the Mac is booting up it will allow you to select a hard drive. I did this and luckily I was able to select my main hard drive and the computer booted up, but once I had logged in and was at the desktop screen the computer was very slow and when I tryed to open Finder I was once again given the spinning wheel. The wheel continued to spin for a solid 10 minutes and the screen finally went black with just the spinning wheel. So I once again decided to turn off the computer by holding the power button. I have gone and talked to 2 different tech support stores and both are pretty sure that the hard drive is failing, which I pretty much already knew, and that it is going to be around 75$ to recover the lost information and another 125$ to install a brand new hard drive.

However, I have been using Time Machine to back up my computer but I have not backed it up since August 23rd which was about 2 weeks ago. So I would lose all the files that I have made in the last 2 weeks basically and since I started school I have done quite a bit of work and so I do not want to lost the files.

What solutions can you come up with that I could use to possibly recover my files?

Any and all help is greatly appreciated.