Yes you are putting things together wrong. Muspellheim and Niflheim are worlds or other realms from Midgard (our world). While they are important to the creation of Midgard, the sun and moon are separate and were personified as Sól and Máni. From the Poetic Edda, in Völuspá, we read this verse:

I remember yet the giants of yore, Who gave me bread in the days gone by; Nine worlds I knew, the nine in the tree With mighty roots beneath the mold.

Then for verse 4 and 5:

Then Bur's sons lifted the level land, Mithgarth the mighty there they made; The sun from the south warmed the stones of earth, And green was the ground with growing leeks.

The sun, the sister of the moon, from the south Her right hand cast over heaven's rim; No knowledge she had where her home should be, The moon knew not what might was his, The stars knew not where their stations were."

So what do we learn here? Well at first the volva recounts that she knows of nine worlds on the tree (Yggdrasil) when she comes to Midgard she talks of how the sun warmed the earth. But in the next line the sun is "the sister of the moon" and doesn't know where her home is, much like the moon. From this we can infer that the sun and moon are separate entities from the other nine worlds.

And as you state in the question, yes, Sól and Máni get eaten by the wolves Sköll and Hati respectively. This as well as their origin is detailed in Gylfaginning of the Prose Edda in verses 11 and 12:

Then said Gangleri: "How does he govern the course of the sun or of the moon?" Hárr answered: "A certain man was named Mundilfari, who had two children; they were so fair and comely that he called his son Moon, and his daughter Sun, and wedded her to the man called Glenr. But the gods were incensed at that insolence, and took the brother and sister, and set them up in the heavens; they caused Sun to drive those horses that drew the chariot of the sun, which the gods had fashioned, for the world's illumination, from that glowing stuff which flew out of Múspellheim. Those horses are called thus: Early-Wake and All-Strong; and under the shoulders of the horses the gods set two wind-bags to cool them, but in some records that is called 'iron-coolness.' Moon steers the course of the moon, and determines its waxing and waning. He took from the earth-two children, called Bil and Hjúki, they that went from the well called Byrgir, bearing on their shoulders the cask called Sægr, and the pole Simul. Their father is named Vidfinnr. These children follow Moon, as may be seen from the earth."

Then said Gangleri: "The sun fares swiftly, and almost as if she were afraid: she could not hasten her course any the more if she feared her destruction." Then Hárr made answer: "It is no marvel that she hastens furiously: close cometh he that seeks her, and she has no escape save to run away." Then said Gangleri: "Who is he that causes her this disquiet?" Hárr replied: "It is two wolves; and he that runs after her is called Sköll; she fears him, and he shall take her. But he that leaps before her is called Hati Hródvitnisson. He is eager to seize the moon; and so it must be."

Not only will they be eaten by two wolves during Ragnarok but there is an origin story where they have a father. This claim of having a father is also backed up in the Poetic Edda in Vafþrúðnismál when the jotunn Vafþrúðnir recounts to Odin the parentage of Sól and Máni being the children of Mundilfari. Therefore I would argue that there are stronger claims to a personified version of the sun and moon than the sun and moon being other realms (Muspellheim and Niflheim) which I imagine you are interpreting as celestial objects when a better way to conceptualize them as alternate dimensions.

Source: The Poetic Edda, by Henry Adams Bellows, [1936]; The Prose Edda of Snorri Sturlson, Translated by Arthur Gilchrist Brodeur [1916]