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.
The Elvish words for “oak” had very similar forms throughout Tolkien’s life. In the Qenya Lexicon of the 1910s, it was ᴱQ. nor(ne) from the root ᴱ√NOŘO, which Tolkien marked with a “?” (QL/67). Its Gnomish cognate G. dorna (GL/30) hints that the true form of this early root might have been ✱ᴱ√NDOÐO or something similar. In the 1930s the root became ᴹ√DORON with derivatives ᴹQ. norno/N. doron “oak” (Ety/DÓRON). This somewhat surprising derivation was explained in the Outline of Phonology from the early 1950s, where Tolkien said:
> n ... also appears occasionally as product of [initial] d, instead of l, by assimilation to succeeding nasals, as in dorno > norno (PE19/80).
In etymological notes from the later 1950s or early 1960s, Tolkien gave the root √DOR “hard, tough”, that in Quenya was preserved only in the word “oak” and in ndŏr > nŏr “land” (PE17/181). Its connection to “oak” indicates this is a later iteration of ᴹ√DORON, though the connection to √NDOR “land” was new. A similar root √DORO “dried up, hard, unyielding” appeared in the contemporaneous Quendi and Eldar essay from around 1959-1960, again connected to √NDOR but without mention of “oak” (WJ/413). In both sets of notes, √DOR has the Sindarin derivative dorn “tough, stiff, thrawn, obdurate”, also used as another name for the dwarves.