This question came up in the Online 10-day Retreat Nov 3, 2020, guided by Delson Armstrong. Day 6 was part of a daily 30-minute discussion on the suttas, the Dhamma talk and reflections.
Primarily, the feeling that is experienced after contacting the Nibbāna element, is the cooling that you just mentioned, the relief. That primarily happens in the mind. There are sometimes statements by the Buddha where he talks about the body contacting the Nibbāna element, but essentially that means that the body is also part of the mind as well. It can manifest in some sense as a physical sense of relief. One of the ways that for example bhante Vimalaramsi explains it is that it's like an ocean of suffering that you know have complete relief from. Your shoulders feel so relaxed because the burden has been lifted and laid down. So, in that in that regard it can feel physical, but it primarily happens on a mental level.