If we consider "I met Caroline while I was on my way going to my home", then home is a noun here.
I think you misunderstood my earlier post.
When I suggested thinking of your original sentence as
I met Caroline (while I was) on my way [= going] (to my) home, I meant that you could understand 'while I was on my way' as 'while I was going' and 'home' as 'to my home'.
Consider these four sentences:
1.
I was going to my home.
2.
I was on my way to my home.
3.
I was going home.
4.
I was on my way home.
In [1], the underlined words are a preposition phrase functioning adverbially. They tell us where I am going. If you want to use the word
modify, then the phrase is modifying 'am going'.
In [2], the underlined words are also a preposition phrase functioning adverbially They tell us where I am on my way to. If you want to use the word
modify, then the phrase is modifying 'am in my way', not just 'way'.
In [3] and [4], the underlined single words have exactly the same function as the PPs in [1] and [2]. As they function adverbially, and contain no preposition, I think they are adverbs. Paul's school of grammar considers them prepositions. I wouldn't lose any sleep over this if I were you.