How can I push back diplomatically against an overwhelming amount of tasks?

As a Senior software engineer working at a mid-size tech company, I’m still learning how to properly push back when others Sr SWE & managers or Directors from Web or Mobile team tried to get me to do tasks that do not match my own priority.  As much as I like to be nice and support others, I agree that I can only do so much.  I brought this up with my direct manager and my Director (L7 Senior Manager & L8 Director), and they told me to loop them in when I face overwhelming pressures from other engineers/ cross-functional teams. My direct manager also told me they wants me to be able to focus on big project initiative, and they see that I am on track to be the Tech Lead given my current trending.

While I do appreciate that my boss gives me words of assurance and direction and offered to step in to fend off those pressures during my one-on-one call, I recognize that I would have to be the person who is good on establish priorities and be able to push back on people. I cannot really rely on my boss to do the push-back to fend off the pressure given that with the recent layoff, we are short on staff.

In light of the recent lay off, my boss confirmed that he's busier than before the layoff, and will be less available given the reorganization within the engineering leadership.  My boss is not going to available and won’t be able to step in when facing high pressure from the other managers. Since my company uses Slack as main source of communication, I usually just cc and tag my manager on slack thread so that raises visibility and awareness.

Wanted to get some thoughts and suggestions on "How can I push back diplomatically against an overwhelming amount of tasks"?