The roots √KHAD and √KHAM were in competition for words having to do with “sit” and “seat” for a significant portion of Tolkien’s life. Both roots have antecedants in the Qenya Lexicon of the 1910s, but in that early document the root for “sit” was given as ᴱ√ÐORO or ᴱ√SORO (QL/85), replacing rejected ᴱ√SOŘO [ᴱ√SOÐO] and ᴱ√SODO (QL/85). Of these, the true form was clearly ᴱ√ÐORO given Gnomish derivative G. dorn “seat”, G. doros “throne”, G. dortha- “settle” (GL/30). This root seems to have been abandoned in Tolkien’s later writing, though N. dortha- “dwell, stay” was reassigned to ᴹ√NDOR in The Etymologies of the 1930s (Ety/NDOR).
As for √KHAD and √KHAM, their clearest antecedents in the Qenya Lexicon of the 1910s were ᴱ√HAÐA “cleave, remain” and ᴱ√HAM (QL/39), the latter without gloss but with derivatives having to do with the ground such as ᴱQ. hamba “on the ground” and G. ham “ground” (QL/39; GL/48). Some variant of ᴱ√HAÐA seems to have drifted in the direction of “sit” based on ᴱN. haud “seat” from Early Noldorin word lists of the 1920s (PE13/147, 155).
Tolkien initially used the root ᴹ√KHAD for words having to do with “sit” and “seat” in The Etymologies of the 1930s (EtyAC/KHAM), but it was replaced by ᴹ√KHAM “sit” (Ety/KHAM) and this root seems to have survived for some time, since ᴹ√KHAM “sit down” appeared in the Quenya Verbal System from 1948 (PE22/103). However, at some point Tolkien added a new root ᴹ√KHAM “call to, summon, name by name” to The Etymologies, and in this new entry he said “KHAM sit (replacing KHAD, cancelled)” indicating ᴹ√KHAD was restored (EtyAC/KHAM²). This seems to represent an ongoing vacillation between √KHAD and √KHAM in the 1930s and 40s.
However, √KHAD “sit” appeared in the second version of Tengwesta Qenderinwa written around 1950 (PE18/95), Q. hárar “sit” (not ✱✱hámar) appears in Cirion’s Oath (UT/305), and in late notes on verbs from 1969 Tolkien had ✶khadmā “seat” as the basis for Q. hanw̃a S. haðw (PE22/148). There are no signs of √KHAM “sit” in this period, so it seems Tolkien chose √KHAD for “sit” in the 1950s and 60s.
Neo-Eldarin: I think √KHAD “sit” is the best choice for Neo-Eldarin, since it also lets us use ᴹ√KHAM “call to, summon, name by name” more freely.