You have it right. I'll explain why.
"10 dollars' worth" is a statement of amount. It doesn't just mean "worth 10 dollars", it means "an amount worth 10 dollars". "Two kilos of rice", "a bagful of rice", and "3 dollars' worth of rice", are all the same construction.
Since "10 dollars' worth" is an amount, you can't apply it to a single object. "A vase" already tells you the amount - the amount is one vase. Trying to add another unit of quantity doesn't make sense. If anything,
I bought 10 dollars' worth of a vase
would imply that the whole vase was worth more than 10 dollars and you cut off 10 dollars' worth of it and bought only that! In that case, if you want to state the value of the vase, you should use "I bought a vase worth 10 dollars".
You're right that you do occasionally see it applied to countable nouns without using an article:
Look at that! $15,000 worth of sports car. All mine.
That's treating the countable noun as if it were an uncountable noun. It's used for emphasis for items that are worth a lot of money. This makes logical sense (though I don't know what a grammar book would say about it); the (one, whole) car is an amount of car worth $15,000. It still carries a certain mental image of slicing up cars, though, so it's only ever used in a jocular way!