First, you're probably not paying "full fare" in the standard sense (because if you were, you'd be able to select a seat at no extra charge). You're paying discount economy fares. In this case, you'll often have to pay to reserve a seat in advance. That's how it works in the airline industry these days.

Second, if you're confirmed on the flight, you aren't flying standby. However, without a confirmed seat, you're at somewhat higher risk of being denied boarding if the flight is oversold. Usually agents will go to significant lengths to find volunteers to give up their seats, but occasionally people do get denied boarding involuntarily. If that ever happens, federal law may entitle you to cash compensation. But you're not flying standby, and they're not going to "sell your seat" during your connection.

Finally, if you're commuting weekly with connections, chances are you'll get status soon enough (if you don't already have it). This will entitle you to more complimentary seat options.