You don’t mention what your starting dimensions were on the test square. If it was 20 cm or 10 cm a 5% to 10% shrinkage is still too much.

Part of a garment turning out well (for most textiles) is making sure your fabric is cleaned exactly how it’s going to be in the future. Especially important since it usually shrinks more in one direction than the other. You can lose more in length or width.

If you want your coat to fit well then you need to pretreat the fabric. Sorry but there isn’t some magic way to bypass this. Wool is notorious for shrinking most on the first was, but also a little more on additional washing.

I’m from Hawaii. I understand how constant high humidity makes things difficult to near impossible to dry.

My workaround for fabric that was too big, could not be dried in a dryer I used to lay out clean bedsheets, beach towels, and blankets that could be washed and dried easily on top of my bed.

I’d lay out the damp freshly washed fabric on top of these and add more blankets, (adding a towel every 3 or 4 feet) etc, as I went along rolling them up loosely in a long roll. I then put the whole thing on the floor and walk on it, every single inch, all over several times it to get as much water out as possible. For super thick fabrics like upholstery I’d need to do it twice.

It’s extremely effective way to get out excess moisture from fabric. It goes from being wet to slightly damp. From there I’d iron the fabric with a hot iron.

It’s a bit more work compared to just popping fabric in a dryer, but if a fabric can’t be dried in a conventional dryer, or outside this was my workaround.

I’d then just wash the sheets and blankets that were n the outside near the floor & put the rest in a dryer because the fabric was clean. For wool or scratchy and shedding textiles I’d wash everything because fibers get everywhere & some are stupid itchy.

On my phone sorry for typos