unless I count "many a" as 2 syllables (MA-nya) and curious as 2 (CU-rious).
This doesn't sound too unnatural to me. (Disclaimer: I speak with a British accent, which Poe didn't.) Let's check how it fits with the meter of the rest of the first verse of the poem:
Once upon a midnight dreary, while I pondered, weak and weary,
Over many a quaint and curious volume of forgotten lore—
While I nodded, nearly napping, suddenly there came a tapping,
As of some one gently rapping, rapping at my chamber door.
"'Tis some visiter," I muttered, "tapping at my chamber door—
Only this and nothing more."
(This is entirely based on my own reading of the poem, and I'm no expert in scansion. I have heard recitations of the poem from several more experienced readers, but I don't remember how much their scansion differed from mine.) Bold syllables are definitely stressed; italic ones are weakly stressed and perhaps it's arguable that they're stressed at all.
If we count both bold and italic syllables as stressed, then every line contains eight trochaic feet, when "many a" and "curious" each count as two syllables. That would make it trochaic octameter, an analysis with which Wikipedia - citing Kopley and Hayes, "Two verse masterworks: 'The Raven' and 'Ulalume'" - agrees. Wikipedia also says - citing Sova, Edgar Allan Poe: A to Z - that Poe himself said it's a mixture of octameter acatalectic, heptameter catalectic, and tetrameter catalectic.
But the distinction is academic when it comes to the particular line you're talking about. Either way, that line has seven or eight feet, and the rhythm is every other syllable being stressed. That requires "man / ya / quaint / and / cur / ious / vol / ume" rather than "man / y / a / quaint / and / cur / i / ous / vol / ume". In short, the answer to your final sentence is yes, it's correct to do so.