How to handle lots of responsibility and avoid getting stretched too thin?

I recently took a job as a Lead Engineer at a company with a pretty high operational tempo. I am now suddenly responsible for a lot of things and I just end up dropping stuff on the floor trying to complete everything. I am still an IC so I dont have any real delegation power.

How do you guys deal with having 12 different things that all have to get done at once, while monitoring these 4 other use cases, and while also making sure various things like compliance and other controls are adhered to for the team?

I dont have the ability to go to my manager and just ask "What is my priority", because they will say "You are the Tech Lead, thats your job. You have to exercise good judgement."

So what are your coping strategies? I am clearly getting stretched too thin and its causing pain in my org.