As you reject any word pairs in which one or both words also have a meaning that is different from the other word, I doubt you will find any examples.
As given in the comments,
total as verbs meaning
to add up the separate values are synonyms, but they are probably normally used in different contexts - and you seem to reject them on that basis.
Likewise, most (if not all) synonym pairs that will be proposed will contain words that can or will be more appropriate in some situations than the other word in the pair, and sometimes one or both will also have different meanings altogether.
If there were any such pairs as you are looking for, I suspect one word will supersede the other: one will become less common, possibly obsolete since there is little use in having two words without any meaningful difference (compare
three times and
thrice. I only hear
thrice used on a regular basis in Indian English.)
It is also possible that one of the words changes meaning to use both words in a meaningful way in the language.
Think of what happened to "boeuf" being introduced in English, being originally the exact French synonym for "cow". It was used in English (as "beef") to mean exclusively "the meat of the cow".
Maybe "for example" and "e.g." fit your definitions, although they are not single words but fixed expressions representing an idea. Actually, "e.g." is usually pronounced as "for example".
Thank you Vilmar for pointing me at botany! Yes, in the names of plants we do find absolute synonyms, and not only in obscure areas...
Eggplant an aubergine are exactly the same vegetable (and the name brinjol or brinjal is used in Indian English).
Similarly, there are spices with several names (curcuma, yellowroot and turmeric are all the same).