I cannot get the text that I want tabbed over, not necessarily centered, to stay saved in the document. Every time I open the document again it has moved over to left margin.
Use a “centered” tabstop. It looks like an upside down T. Find it in the top left corner of the ruler, or set the paragraph itself to alignment “center”. In general, a lot of needed information is missing: Which operating system, which version of LibreOffice, which document format, how do you try to get centered text.
What version of LO are you on and what format are you saving your file in? I’m running Version: 5.3.3.2 on a Linux Mint 18.1 build and had no issues saving with tab stops and any formatting populating properly in both .odt and .docx formats.