REALITY CHECK - Need more info (how many months is the loan, 60 or 72 - or longer?)
When the dealer says they'll, "Pay off the car" that seldom means they are giving you $16k in trade. It means they are giving you the 13 (or whatever) and rolling the remaining difference (so up to 3 grand) into your new loan. This means you are handing them money on the front end AND on the back end. What you're probably really paying is the $26k for the new car, plus the $3k you still owe on your old car for a loan total of $29k (plus TTL).
Unless you have it IN WRITING that they have valued your trade for sixteen grand and would write you a check for that amount.
Look at your paperwork VERY closely. If they aren't doing that they've messed with the interest rate or the loan period to get you the "payments you want", but are still getting their money out of it.
We can't really make an informed decision without all the particulars of the proposed loan agreement, but if it seems to be working out in your favor it rarely actually is.
-Also, rereading your initial post I get the impression the dealer has not physically inspected your car yet (you stated you "told" them the facts on your car)? That would mean their appraisal is conditional and I'd be curious if they don't try and knock it down a bit once they actually put their eyes on it.
In case it's not obvious - I don't trust car dealers.
That being said, according to my trusty financial calculator, 427/mo for 72 months @4.5% does work out to the final price you've stated for the new car, so maybe it's legit and they really are giving you the payoff amount in trade on your old one. In which case, that seems like a pretty good deal.