Q. Is it true that one shouldn’t discard old Jewish newspapers because sometimes they contain God’s name?
When the Hasmoneans defeated the enemy they ordained that the name of God should be mentioned even in bonds, e.g. “In the year … of Yochanan, high priest to the Most High God”, but when the sages heard of this they said, “Tomorrow the man will pay his debt and the bond will be thrown on a dunghill,” and they stopped them. This is why anything bearing the name of God or with a Torah content must not be discarded but given respectful burial in a Jewish cemetery.
One might express a view that the Jewish press should contain more Torah material, not less, but it should certainly take greater care about not using the Hebrew name of God, even in illustrations, and readers should cut out any Torah material and add it to the Shemot (“Divine names”) box in your local synagogue. This box is regularly emptied and the contents given to the Chevra Kadisha.