The amber/yellow LED indicates that the MiFi is charging with the power off. You can also see the Amber/Yellow LED when you power up/down the device. It does not normally indicate a problem with the device.
However, if your MiFi is shutting down unexpectantly then there might be an issue with the battery or the charger. I would suggest removing the battery for a while and running purely off the AC/Wall adapter. See if you can reproduce the problem. You can also try alternative USB cable charger's to compare and contrast, any standard Micro USB charger should work with the MiFi.
If you can reproduce the problem with the wall charger only then your MiFi may be defective and should be replaced under warranty. If you cannot reproduce the problem then your battery is probally bad and should be replaced under warranty.
Let us know how it goes and we can suggest more troubleshooting steps from there.
I am experiencing this same problem, but am confused by your suggestion to pull the battery and run on AC power exclusively. The device indicates I have full battery power (we always keep it plugged in) so I suspect the battery charger is not the problem. If I pull the battery and leave it plugged in, it goes to the amber light almost immediately. I thought a battery had to be used. So if I have full battery power, then pull the battery and it immediately goes to the amber light, what does that tell us? Usually the device will stay powered on for hours before it goes to the amber light (with the battery, and plugged into the wall).
MiFi's can only hold a charge for 3-4 hours with normal use. If your MiFi is powering down after hours then it is simply running out of power. Doesn't sound like there is anything to worry about based on what you posted so far.
The purpose of the steps above was to isolate the problem to either the battery, the battery contacts on the device or the charging cable. If you think your battery and cable are good then the only thing left to replace is the device itself. Contact VZW to see if you are still covered under warranty and they will send you a replacement.
The device is always plugged into the wall (my office simply uses it as a "dedicated" hotspot). Therefore, the battery is always fully charged (and the indicators show a full charge). The device worked fine for a long time. I read on other threads that the documentation/statements about running this device without the battery was wrong (that's what was causing my confusion; via other threads I learned that as suspected, the battery IS required at all times). We are going to try to unplug the device, let the battery drain entirely, then recharge it. If that doesnt; do the trick, guess I'll return it. If you have any other thoughts regarding why the device would power down intermittently, when it is plugged into an outlet, and the battery has a full charge, please let me know.
I'd try calling it into VZW and take advantage of the 1 year warranty that comes with these devices. If you are still covered and explain your symptoms to a rep they should be willing to send you a replacement with no hassle.
If your device is no longer under warranty then you will have to ride it out for a few months until you are close to your contract expiration. When its time then cancel or upgrade to start a new contract and pick up the newer 4620L. That technique would be much cheaper than purchasing a replacement MiFi outside of a contract.