One potential problem with this plan is that you may still owe more on the cars than they're actually worth.
The standard "price guide" for cars is the Kelley Blue Book, and there are several versions. In general the most you can reasonably expect to get for the cars if you trade them in is the "wholesale" price, which is what they're worth to a dealer (the amount they'd expect to pay for them at auction or from another dealer). This is likely to be less than the amount remaining on the loan; it may be significantly less than the amount remaining on the loan.
It may be worth trying anyway. There's a lot of mark-up in aftermarket accessories, so if the dealer is soaking you on pinstriping or rustproofing, they can afford to give you a bit more than wholesale on your trade-ins and still make a nice profit.
If even after this you're still "underwater" on your existing loans, it MAY work anyway; you can "roll over" a certain amount of debt from your previous vehicles onto the new one. This might not help a lot on the "can't afford two loans" front, because, well, consider the following example.
You bought both your existing cars for $20,000 each. You still owe 7k on one and 14k on the other. They're worth respectively 5k and 10k as trade-ins, with the dealer being as generous as he feels like being. This means you've got 6k in debt that's not going to be covered by the trade in.
A lender might be willing to allow you to roll over 20% of the new loan, meaning to cover your 6k you're going to need to buy a $30,000 vehicle, but the loan is going to be for $36,000. That's not a LOT less than the combined total of the two $20,000 loans you had, so the payment for the one loan is not going to be much less than the payments on the two $20,000 loans.
It's possible your situation is different, and this might actually work out for you, especially if you got crummy terms on the first loans and you can do better now (maybe your credit score has improved a lot, or something). It would certainly be worth at least talking to a dealer to see if it's a viable option. One thing that could help would be if there are rebates on the new car ... you could use those to immediately cover at least part of what you still owe on the old cars, meaning you'd be financing less and lowering your payment.