I suspect that you have damaged some pins on the Arduino.
Use a good quality voltmeter to measure the output of the ADXL335 chip. I am sure that it survived.
I am guessing that with the silk screen side up the board will be reading something like the following:
But, First measure the supply voltage -- it should be 3.3V (Between ground and VCC)
Since there is not a lot of room on the sensor breakout position the probes wherever necessary so that you don't short things ... But do measure the voltage at the board. Make sure that your ground wire actually connects -- you could have floating voltages.
Then measure the outputs...
1.8 - 2.2V for Z
about 1.2 -1.5 V for X and Y
The main thing is that there is approximately that difference -- don't get hung up on exact numbers. If the sensor is not positioned level it will be different....
Once you know that the voltages are correct then you can move the leads to three new analog pins and try those pins.
YOU DO NOT NEED TO USE THE External Reference to start -- just see that you get something other than 1024...
Then once you know it is correct (And you have the command in your script) plug in the AREF and you should get about:
Z -- around 650 ~ more or less
X&Y about 520 ~ more or less
Don't have a voltmeter? Get one $60 US/CDN gets one with Frequency Measurement and all that. I have the equivalent of the UT60E -- I got mine from Steren.
This is basic troubleshooting.
The Compiler on this site is your friend. Use it -- it's simple and there are LOTS of examples to start out.
I have been at this over 40 years and designed math and communications systems. I was not the least bit embarrassed to start with the simpler scripts and learn from the people who already tried this stuff. The examples are for the most part excellent!
That is the best advice I can give.
PS: One more thing... Maybe you ran a PINMODE script and a digital write that set the pull up resistors high. There are a lot of scripts where the anlog pins are used to "do something digital" or they will condition pins to provide power and ground to switch devices on or off... I avoid them, or If I do want to try the example, I convert them to "normal" digital pins and I just use normal continuous power. Get fancy after you have some experience. See the reference area for a better explanation!