It's a bit too outdated, I think. The word "Kabinett" is usually only in use for the political Cabinet (of ministers). (Or for a kind of wine. :) )

"Der Schrank" is, in my book, first and foremost a wardrobe (two metres high, for clothes). However, the word is just as well used for, well, any furniture with a door on it, really. If somebody asks you to get them a towel "aus dem Schrank", they'd have to point you to the wardrobe / cabinet / sideboard / kitchen cupboard in question, because you wouldn't know what kind of Schrank they're referring to. (cf. Schrankwand = wall system)

A number of prepositions, including in, can take either dative or accusative case: the dative case to indicate a location, the accusative case to indicate the destination of motion.

Here, the towels are "in" the cabinet (lying there motionlessly: the cabinet is their location); they are not "into" the cabinet (destination of motion).

Thus the German has im Schrank with dative, and not in den Schrank with accusative.

In practice, I would say that im is essentially always contracted when it means “in the”. It has to be two words when it means “in that ...”.

What was the entire sentence that you wrote?

Did you get the gender of Schrank correct? It's a masculine word, so if you had, for example, Ich kaufe eine Schrank, that would be incorrect (using the feminine form eine with the masculine Schrank).

Yes, "(Hand-)Tücher" is the plural of "(Hand-)Tuch".

Here's a short discussion about umlauts in plurals.

If 'Handtüche' is 'towel'

Handtücher is the plural; one of them is a Handtuch.

Handtüche doesn't exist.

what is a hand towel?

Theoretically a Händehandtuch.

But we don't usually use that precise a word.

If it hangs in the bathroom next to a sink, it's simply a Handtuch -- you can dry your hands with it or your face. If it's a bit smaller, it might be a Gästehandtuch "guest towel" since those are often smaller than the ones for family members. (No idea why.)

A bigger one for drying off your body after a shower would also often just be a Handtuch. (Though there's a specific word Badelaken as well.)

So you might talk about ein kleines Handtuch und ein großes Handtuch "a small towel and a big towel" if you want a hand towel and a body towel.