Yep, with the super thin grounds filter, I see your problem. I use this, which has a much wider filter and no such problem: https://www.amazon.com/gp/product/B01N0CLJLH/ref=ppx_yo_dt_b_search_asin_title?ie=UTF8&psc=1

I don't drink much coffee and had never heard of a Toddy! Sounds like that's the way to go. My only minor quibble is that I'm trying to drive plastic out of my foodchain as much as possible, and the toddy is one huge plastic jug. The metal filter on the Bialetti I linked is held together by a few plastic strips, but a lot less of it. That said, if I were buying a new system, I'd try the toddy out anyway. As it is I just don't drink much cold brew regardless these days