Neighbours' conifers are brushing against my house. Should I offer to pay to cut them down?

goldfinches

I'm chiming in to say all of the above sounds jolly helpful to me especially if I try to consider it from your neighbour's perspective rather than yours IYSWIM.
My other paragraph is about social anxiety. It may be helpful to you to take an approach that minimises your time thinking about introducing yourself to your neighbours, which clearly causes you to worry dreadfully, and simply to decide to get it over with so that you can stop that particular worry.
You might find that telling yourself that this will be over in five minutes time and that you will have accomplished this task will help you to cope with knocking on their door.
Then however it goes you won't have to do it for the first time ever again.
There are, of course, many other things which immediately present themselves as worries but this particular one is now gone never to return. 
Sometimes this strategy, or your own variation on it, can be helpful with coping.
Best of luck whatever happens, goldfinches.