I have a Toshiba laptop from 2013, that I use for testing purposes.
I made a fresh installation of Ubuntu 18.04.1 LTS, made it up to date with
sudo apt update
sudo apt full-upgrade
and rebooted to have the current kernel version and everything up to date. Then I unplugged the power and let it run (most of the time playing music videos) until it started to warn about low battery. I watched it closely and at about 5% remaining power it shut down automatically and gracefully.
I plugged in the power connection again, let it charge for slightly more than one minute (so that there should be enough power for the booting process). You may need 5 minutes or more if the battery is getting tired.
Then I booted and it was successful, except that I had to turn off airplane mode for wifi. I tested a few things and they seem to work correctly.
I shutdown and booted from a live drive and checked the file system,
sudo e2fsck -f /dev/sda1
Everything looks good, there was no complaint.
Ubuntu 18.04.1 LTS fully up to date (today, 2019-01-26) can shutdown gracefully, when the battery is getting low in my Toshiba.
I don't think dual booting should make any difference, because the same system will be installed, although the partition number of the root partition will be different.
If Ubuntu shuts down gracefully, when you initiate it manually, the automatic shutdown should work too, unless you have modified some setting, or if the battery supervisor does not work with the computer's hardware.
I suggest that you check that the battery supervisor works with your computer's hardware.