I created drawing template, by adding tables and removing sheet format.
Tables were anchored to appriopriate points.
Then I made sheet formats with corresponding anchors.
When I start new drawing I am asked to choose sheet format. As long as I pick format I used when creating drawing template it is fine, but as I pick any other format, then tables do not anchor properly unless I enter Sheet Properties dialog window and reload the sheet format.
It sounds like your sheet format was saved after your drawing template was saved. Drawing templates use the sheet format that existed at the time the template was saved, and will not automatically reflect changes to the sheet format. Open your drawing template (or start a new drawing with it), go to Sheet Properties, Reload the sheet format, then save the Drawing Template (or save as drawing template if you opened a new drawing).