Yes, Skye is a very outdoors place and the west is where the rain comes. Sigh. It is why my walking group bases itself near Inverness. If it's clear in the west we head to the coast, but if it's raining we have options. I did a photo workshop there for a week and I think I have one day of sunny pictures. I learned to appreciate the up close pictures and the muted colors. They really popped on a misty day. When my sister and I made our first trip way back in the 70's we did the walk up to see Loch Coruisk and got drenched because it rained all the way back to Portree and we couldn't find the bus. Someone finally took pity on us and gave us a lift. They didn't seem to mind that their car seats were soaked.
I assume you've been to Dunvegan and the museum of Island Life. I know both will still have outdoor bits, but that's life in the west.
Can you escape by going east? (Assuming you have a car.) Look at the weather to see if that would make sense. Inverness is 2.5 hours from Portree. You could get up to Loch Ness. Or go all the way and check out Culloden. There is always shopping too in Inverness. :) There is a great knit shop in Beauly. Of course you would drive right past Eilean Donan Castle. If you just plain need to escape, and don't have a car, well, maybe the train to Inverness and back?
I hope you have good water proof.
Pam