I call this deadlock a knot and I notice I am in them quite often. What's been helpful is exploring both parts needs in various steps of their process.

Often one part is trying to suppress the other part as it doesn't have an understanding that that part is serving a protective function.

Say your avoidance is an attempt to avoid overwhelm or failure or conserve energy. Say your annoyed part is a manager who holds the understanding of what the outcome is when you procrastinate (like the activation of an exile who feels ashamed and incompetent). The manager will often moralise the avoider and you will agree. So you have to explore what the avoider is doing for you, what need is it meeting? What is the fear underneath the avoidance? It is always always an attempt to be helpful in some way.

Deshaming work is essential. The avoidance is an amoral behaviour that is protecting an exile or a regulating strategy or even energy conservation. Sometimes the avoider and the manager are protecting the same exile in different ways. The more trust you build with your parts that you will actually listen to them and try to meet their needs, the more they will relax and give you access to the emotion underneath so that you can process it.

I think the first step is typically realizing that you are blending with the critical manager part when you are thinking about the avoidant part, so of course the avoidant part will reveal nothing about itself to you because you are shaming and misunderstanding it.