What is the name of the relationship between two companies that have the same principals where the new entity is operating to avoid debt

I assume that you contracted with the company, not its individual owners, and that the company is a separate legal entity, such as a coproration or llc.  As a general rule, the owners of an entity, such as shreholders of a corporation, are not personally liable for the tntity's obligations, and separate entities are not responsible for each other's obligations even if they have the same owners.  In rare instances you may be able to "pierce the veil" of an entity to hold its owners responsbible for the entitiy's obligations, or hold a successor entity responsible for its predecessor's obligations, but not often.  The rules for doing so are too complex and voluminous to go into here, but you would certainly need to show that there was some impropriety in the way that the parties (either owner and entity or predecessor and successor) conducted business with you which warrants making an exception to the normal rules.