Dave,
You are correct that it is difficult to make a team's priority the top priority when people are worn thin with other responsibilities. One suggestion is to block time on your calendar and put your DND button on your phone for 30 minutes, a few times a week. It is much more daunting to tell yourself that you need to work on the project for "several hours" because often times, we don't have several hours to give. Scheduling yourself for shorter sessions will help make the team's goal a priority and management. Give it a try and let me know if it works for you.
Thanks,
Jamie