I've taken my dog in the Discovery trail and the multi-purpose trail up in the mountains. What other options are there?
Robin Hill Park is a nice one, lots of shade and there's a big open area as well.
Peabody creek
Was curious and found this too:
Tubal Cain Mine has always been one of my favorite (on leash) dog friendly trails! Super pretty in the spring.
Google Maps is saying that's 2 hours away. Is that right?
That sounds about right. If you're looking like IN Port Angeles, a few to look at are the Waterfront trail, the trails/power line cut above the college, Peabody Creek Trail, or the DNR roads between black diamond and Hurricane Ridge Road.
Close in Little River is dog friendly until you hit the park boundary. Verne Samuelson trail feels like you are right in town then quickly in nature
Lots of people walk their dogs in the trails behind the Elwha Fish Hatchery but it's mostly off leash, so you'll have to be OK with that
Striped peak of course.
The beach at the mouth of the Elwha is a lovely walk as well
Mt Pleasant is on DNR and therefore dog/bike/quad friendly. It's logging roads and not a trail per se but it's easy to stay out of the way.
Pets are permitted on the Spruce Railroad Trail ... a bit of a drive, but pretty ;)