Do I need to register my completely online business as foreign LLC where I live if originally incorporated in another state?

There is nothing like a 100% online business even if you sell online. It is important to understand that where you are physical, specially in NY, you may need a license, and if you use a name other than your legal name, you must obtain a DBA registration. Additionally, if you are located in NY but you sell online to customers in other states, there is now a legal threshold after a certain amount of transactions or sales that state will consider you as "doing business" in that state. If you don't deal with that you could face tax issues. Now, whether you need to incorporate that is a decision that involves several analysis. However, keep in mind that you have a physical presence you probably need to. Finally, if you meet that threshold, mentioned above, if a state consider you afe "doing business" you could be required to register or get an application for authority.