PIC Time
You mentioned your dream jobs, so go to their website to see what they want in order to be picked up. Some airlines require ME turbine part 121. So keep that in mind. It will affect your overall decision.
You mention that you don't want to go to a regional and I'm not sure what the reason for that is, but if you don't have any 121 experience, sometimes you have to go that route. The majors like to see part 121 time because it helps them to see that you can "get along with passengers" and the like. Yeah, boxes don't complain, but people do, so they want somebody that is a people person.