What you are describing compares to a lot of the PNW, especially in winter :-) Moss is probably the easiest plant to grow in this area! Acidity should not be a major concern unless it is radically skewed to the lower numbers -- a simple home pH test (use distilled water or a metal meter) should verify that. Most shrubs (perhaps the majority of woody plants) prefer a somewhat acidic soil anyway.

What could be of concern is drainage. Moss is often an indicator of compacted soils, reflecting poor drainage. And while your soil may feel squishy, especially when wet, it may also be a sign that drainage is not good. And few plants are happy under poor drainage conditions. Adding coarse organic matter to the entire planting area before adding a lot more plants may be of value, as this will help to loosen compacted soils and improve drainage. Raising or berming the planting area can also be helpful.

Some moss species grown on non-acid substrates. Here there are species that are characteristic of places like concrete parking barriers, due to them being pretty much confined to non-acidic growing sites.

Mosses and other bryophytes have been said to pile up on the trunks of the native bigleaf maple due to the tree being rich in calcium.

Mosses, like ferns need a film of water to reproduce, tend therefore to appear on compacted or dense surfaces.

would not the first question be ... what if any problems are you having with your shrubs ....

'fixing' something that is not really a problem.. may lead to more problems .. rather than being a 'fix' ...

if there isnt a problem .. dont love them to death by trying to fix what is not a problem ...

otherwise .... drainage is the issue ... and just like the peeps with clay .... you plant on a berm.. so the plant can keep its roots as dry as it needs while getting 'established' .. while it puts its roots where it is pleasing to the plant ...

i have pure sand.. drainage is not an issue... and i have moss everywhere .... water is not an issue for my moss ... its green in spring and fall .. when its cool and damp .. and looks like brown death in summer ...

no problems with conifers .. shrubs .. trees or hosta ....

and.. shall we presume that this has nothing to do with a septic field or tank??? if you are on septic.. you might have giant problems ...

good luck

ken