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Late to this, but on Lk. Washington there are good ramps with docks at Stan Sayres hydroplane pits on the west side south of the I-90 floating bridge.
Also another launch with dock at Sand Point further north on the lake, on the west shoreline a couple miles north of the 520 floating bridge.

Best launch on Lake Samammish is at the state park at the south end of the lake. It also has docks.

Your 13' will be fine on these lakes on MOST summer days. Other seasons - watch the weather closely like anywhere else.

On Puget Sound in a 13' boat of any kind watch the weather very closely all year around. Also very strong tidal currents in areas that when against the wind direction steep seas will develop very quickly and seemingly out of nowhere. The "Narrows" just outside Gig Harbor (on your way to Fox Island) are notorious for this. Get a tide and current book and pick your days and times well.

Nothing (including the ramps above and most other ramps) is free here. Sorry.