After I update the High Sierra, my external hard drive can not act perfectly and is read only!

https://support.apple.com/en-us/HT203538

I had this problem also and the permissions were still grayed out after unlocking. After searching for answers it was clear that my external hard drive, being formatted Mac OS Extended (Journaled) should still be read-write under High Sierra. I attempted to run disk utility First Aid a number of times but received error messages. Now I seem to have solved the problem by unmounting and remounting the disk with disk utility.

I suggest you try running First Aid and then unmounting and remounting the disk.

Time Machine is not backing up to my external drive right now and says I need to use Disk Utility to repair it. Disk Utility can't repair it because it's mounted. I can't unmount it because it is in use. Any suggestions?

If it is formatted as NYFS, you need third party wcrom Tuxera or Paragon. The version for your older OS is not compatible with High Sierra and needs to be updated.

I had this same problem and as you stated i tried Tuxera and it worked. at some point i will copy the files and reformatt the drive