I second Squam Lake. It’s quiet, minimally developed, and it has nice islands to explore in the middle of the lake. You may see loons and eagles. The boat launch is in Holderness on the creek connecting to Little Squam.
I’m not sure where on Winnepesaukee you were considering. There’s a lot of put-ins. The whole place tends to be overcrowded on summer weekends, but a weekday in September shouldn’t be that bad. If you’re looking for recs there, I’d start with Moultonborough Bay. You can put in at the Lee’s Mill boat launch at the end of the bay where it’s shallow and there’s no wakes or jet skis allowed. Or you could launch at Pier 19 in Tuftonboro and explore the islands near the mouth of Moultonborough Bay.
I’ve never been on Sunapee. I’ve driven by it countless times, and it just never looked that great for paddling. So I can’t give you any recs there.
I don’t know which direction you’re coming from, but if you’re coming from the south (Boston area) you might also consider some of the southern NH spots like Pawtuckaway and Massabesic. Or Great Bay for something different entirely (mind the tides though!)