Before Exo has to get down to looking at the emulator code lets do some trobleshooting to rule out a few things.
First off which moto G do you have processor/android version would help.
--------------------------- system test 1
I am going to offer this help based off of android version 4.4.4 kitkat since that is what my tablet runs currently so I know it the best it may or may not work the same on lollipop.
1. Enable developer options. (Should be tapping the android version number some large number of times but may vary from manufacturer, google is your friend here.)
2. Under defeloper options there should be an input section with 2 items (show touches and show pointer location) enable both.
3. You should now see a visual representation of your physical touch screen presses.
4. Run DrasticDS and see if where you think you are touching is where your phone thinks you are touching while under the load of this emulator.
-------------------------------------- drastic test 2
Run a different rom entirely. One of the professor layton games would be good for making sure where you press is what is pressed in the emulator. If amdifferent game works and you ahould test a few the the issue is either game specific or you have a bad rom of LoZ:ST
------------------------------------- Other info
If you are already on lollipop then it may also be a driver issue with qualcomm's drivers here is a nice pose in general about that
http://www.drastic-ds.com/viewtopic.php?f=5&t=2465
If you are running a cracked version of Drastic (go buy a copy) then we know Exo does something to mess it up and this may be on purpose in that case the only help is to buy a legitimate copy. Also there is a nice beta program to help test new versions so if you have a legit copy you have no reason not to update to the latest.
This is all the help I can offer.