"Either way" means that whatever just happened doesn't make a difference to the upcoming event. "I thought he would be alright. Either way, we have to carry on as if nothing changed." "Either way" is used to express that whatever just happened doesn't change any future plans. This doesn't mean, however, that it is unimportant. It just means that it doesn't change any future actions. "I didn't have any motivation, but I had to go to work either way." Using your example, "Either way, he was gone and I was here, carrying the responsibility" means that you still have to carry on your responsibilities, even though he was gone. Hope this helps!

It means irrespective of which of the two alternatives is correct (which you know from the context), the result is the same. For example, in your sentence, it could be that your colleague John said he had to leave for an emergency, but you are not convinced. However - "Either way, he was gone and I was here, carrying the responsibility."