The great thing about our beaches is that even the more popular spots on a holiday weekend won't feel anywhere near as crowded as other beach towns often do. There are over 30 public beach access points and several beach park areas that include free access with extra parking.
For a more secluded location I would recommend 4 different locations. First is the beach area inside Fort Clinch State park. It's at the northern most end of beach and can be quite secluded. Please note that this is one beach area that is pet-free meaning no dogs allowed on that section of beach. There is a pier that has a stretch of large rocks below it. We have found these rocks to be a great place to search for 'treasures' such as shells, sand dollars and shark teeth (our favorite). There is ample parking, but you would need to pay the state park entrance fee ($5 I believe). Also, another advantage is that you'd be in the state park and could also partake in the beautiful trails or tour the fort.
The next place I recommend is North Beach park. It is definitely more secluded because it's not near any hotels or attractions. The only things nearby are houses. The only downfall is there is minimal parking. Still the parking is free and the access is public. North Beach is located on North Fletcher Ave - about 2 miles north of Atlantic Ave.
Towards the south end of the island is the Scott Road beach access. This is just south of the Ritz Carlton resort. There is free parking here as well.
Finally at the south end of the island is South End Beach access which is a couple of miles south of the Amelia Island Plantation.
All of these are fairly secluded areas with free parking (except for the entrance fee to Ft. Clinch state park). Please note though that many of these do not have public restroom facilities, so if those are a must you may want to try a larger beach park such as Main Beach or Peters Point.
The beach areas are all very well-kept and the sand dunes are well preserved. There are board walks to get over the dunes. You will likely find some shells on any stretch of beach, but we're more known for fossilized shark teeth than for shelling. It's kind of exciting to find the teeth for a keepsake! Regardless of what beach spot you choose, I'm sure you have a very nice relaxing trip.