Dogs are permitted in both Avalon and on the Greenway trail. A number of my friends walk their dogs in either place. There is no tick free place. Avalon has a lot of brushy areas, that's the only downside.
Another alternative, is to walk along the sidewalk from the Stony Brook Yacht Club (large public parking lot just opposite the Three Village Inn) down to the pond, and then head up Harbor Rd to Rhododendron Rd. Make a left on Rhododendron, follow it to end and make a left. Continue until you pass a small pond on the right, and turn left. It brings you back out onto Main St just north of the light at Main and 25A. Turn left and walk back to the SBYC past the Mill Pond. It's about 3 miles and a nice walk which will keep the dog out of grassy areas.
Now that most of the students are home for the summer, SBU has a lot of great bike paths for walking.
Going back to the Greeway Trail. there is also a small lot by Ren Tech at Limroy Dr and 25A -- it's just to your left as you turn onto Limroy. This is the actual start of the trail.
Please be kind and remember to pick up after your dog