What you planned to do is perfectly legal. It's called end-on-end ticketing meaning you're combining 2 tickets to get to your destination.
In the fare rules for both DFW-NRT and NRT-SIN, you'll see:
Code:
END-ON-END - -PERMITTED
Not a bad way to get around the 6 days stay.
One word of caution though, since you'll have 2 seperate tickets, if you miss your NRT-SIN because your DFW-NRT flight is late (or miss your NRT-DFW because SIN-NRT is late...) for whatever reason, UA is not responsible for protecting you.