I've been having this problem. After about four or five years of refrigerating my soaps and NEVER ONCE getting soda ash, I have suddenly started to get it. I am not using goats milk so much so I thought that might be a reason. I came on the forum begging for help a while ago. I too was getting white crumbly edges immediately on unmoulding on the bottom corners of the cavity moulds which penetrated into the soap, it wasn't just superficial. Not too much on top. Last time I made soap I followed advice and put the mould on a warm metal tray immediately after pouring, into a slightly warm oven , for several hours and leaving it for at least 48 hours before unmoulding. This seemed to improve the situation though I got more ash on top! I am just embarking on a new batch and this time will try putting cling film on top as well. I am approaching my soaping with some fear and trepidation as I find ash really annoying and depressing after so many years without it. It's not fair!!! The weird thing is that different varieties with the same fundamental recipe are differently affected. My Cinnamon, Honey and Oatmeal I still refrigerate - it comes out fine. My sea salt soaps are fine. The worst is Lemongrass and Poppy Seed followed by Mint and Tea Tree. Lavender, Sandalwood and Orange and Patchouli vary in how bad they are between batches, but aren't usually as bad. My plain soap seems to come out fine, but that alone of all the soap recipes is slightly prone to DOS! I'd love to know why they are all so different. It makes no sense, unless the actual essential oils have some effect on how the soaps are behaving, which is the main thing which differs between each soap variety.
Good luck with your quest. I think trying it in a warm oven might be worth a try.
Sorry, I should have been clearer - if you try the oven warming method, you need to switch the oven on for a while (while you make the soap) on its lowest setting, then switch it OFF when you put the soap in. You don't want it too hot in case it volcanoes. This is something else that so far, has never happen to me, so I don't want it too start.