Arthur still feels like an interloper in Adia's apartment. He's been paying for it in cash, and yard work has been steady enough to do so - it's not like he spends on anything but food and shelter.
The problem, of course, is that he knows what he's looking for now, but has no idea as to where to start. You don't just start knocking on doors and ask "Hello, I was wondering if you had any ancient swords lying around?"
He still feels stuck. The only difference was now, he had a specific task.