Welcome to the forums!
Please correct me if I'm wrong, but from reading your post it appears:
1. Your AC adapter went bad and you replaced it.
2. Your battery is no longer functioning correctly for whatever reason.
3. Running your notebook without the battery and on AC power alone still does not go well, and the notebook will power off fairly quickly.
My questions and suggestions are below.
A. Where did you get the new replacement AC adapter from and can you check the rating under the AC adapter to see if it matches the label under your notebook (possibly in the battery compartment?)?
You may see something like 19V (volts) and 7.9A (amps) for the 150 watt adapter if your system uses it, but I just want to try to make sure you have the right AC adapter.
B. This may not be possible if your battery isn't working well, but it wasn't clear to me if you see any problems when running only on battery power without the AC adapter, so curious how this went if it's even possible.
C. Is your notebook still under warranty? Be sure to check your invoice which may give you a little more time and if you are located in the U.S., you can reach ASUS support through this page if necessary
http://www.service.asus.com/#!consumer/csn9 .
If you don't get better ideas to solve this soon, you may just want to send the unit back for warranty service if possible, or maybe out-of-warranty service if you're no longer covered and want to pursue this route which we can also discuss more if you're interested.
D. If you want to look into it more, search these forums about checking your temperatures, but if you don't have a way to re-apply thermal paste, clean out the fans, or do more extensive troubleshooting, you might just want to have us or somebody else check your unit for you.