Dear Ani,
Thanks for posting your learnings on here!
In this case, your etymology is being lead astray by the nature of Pali compounds.
In a term such as saccādhiṭṭhāna the compound resolves thusly:
sacca + adhiṭṭhāna
truth + foundation
There is in fact no root ādhi here. It is, rather, the prefix adhi, which is lengthened when combined with the final vowel of the preceding element, a feature called “sandhi” in Indic grammar.
Fun fact! Both sacca and adhiṭṭhāna are themselves compounds.
sacca = sat = -ya
truth = real, is + (abstract noun ending), i.e. “realness”.
adhiṭṭhāna = adhi- + (ṭ)ṭhāna
foundation = (prefix implying intensity) + “standing place, situation, position, place”, i.e. “place of firm footing”, i.e. “foundation”.
Be careful, for the final arbiter of the meaning of words must in every case be the context, but sometimes etymology can be handy. Here, the derivation of the full meaning follows quite directly from the etymology, but that is often not the case.
The root of samādhi is in fact the verb dahati, with the primary meaning of “gather” and secondary meaning of “fire”.