I am registered for sales tax in Texas for a small mail order retail business.
Am I obliged to charge and pay sales tax for sales I make which are not part of this business, for example, reselling my own household items on ebay or craigslist?

In theory should every private person making these kind of sales be charging and paying sales tax? I know most people do not.

If you are selling your personal items, Texas proabaly has a casual sale rule regarding sales tax.

However, you better keep your business accounts and your personal accounts separate. If you start comingling funds, if/when you get audited, you will have a lot of explaining to do.