"While my experience in working relationship one-on-one is excellent, it becomes really difficult to manage conflicting priorities when the size of the team pushes 4 or so members." Yes, welcome to the world of herding cats -- and IT cats are the hardest to herd!
So let me suggest that you consider the level of support you NEED from each person on your key issues by using the following continuum:
I need them to LET it happen » » » » » I need them to HELP it happen » » » » » I need them to MAKE it happen.
Then, using the same continuum, consider the level of support you CURRENTLY HAVE from each person on those key issues.
Less confident/practiced IT project managers tend to spend most of their time with the people who are already giving them as much, if not more, support than they need for a given issue and not enough time with the people from whom they need more support than they currently have.
You can substantively increase your impact and influence, without having to engage your executive sponsors in the minutiae, by focusing your attention on those who's support you currently NEED > CURRENTLY HAVE. Then, the conversations with your executive sponsors can focus, instead, on more meaningful updates and progress reports and maximizing executive-level support and visibility for the initiatives under your charge.
I'm happy to talk through some ways the make this happen for you, if you'd like.