Being busy and being effective are very often different animals. If you're busy with stuff that doesn't really matter it's a much less effective use of your time than concentrating on that which does matter.
Your problem isn't finding time to clear your backlog, your problem is building behaviors to manage your pipeline. When you focus on a backlog, you're working the symptom instead of targeting the cause, looking backward instead of forward.
Here's how I help my clients change email behavior -- starting with their own habits, and help others to do the same. If some or all of these are done consistently, you can change the game in your organization.