How to correctly fix a "dirty" NTFS partition without using chkdsk

Try using ntfsfix, which was part of the old ntfsprogs package and is now located in the ntfs-3g package (which you may already have installed).
This is a command line utility which attempts to fix common NTFS problems. It's been a while since I last used it, but it should do its job simply by typing:

sudo ntfsfix /dev/sdb1

It also offers a specific option to clear the "dirty" flag on the partition:

sudo ntfsfix -d /dev/sdb1

which might or might not be required in addition to the first command. Have a look at man ntfsfix for more information.