The basic root for rulership was √KHER for most of Tolkien’s life. The root appeared as ᴱ√HERE “rule, have power” in the Qenya Lexicon of the 1910s, and in this period already had the derivative ᴱQ. heru “lord” and ᴱQ. heri “lady” (QL/40), words that retained the same form and meaning throughout Tolkien’s life. Gnomish derivatives from this period include G. herma “protection, lordship, sway”, G. hermon “lord”, G. hîr “care, anxiety; heed”, and G. hiril “queen†, princess” (GL/49).
In The Etymologies of the 1930s these last two Gnomish words became N. hîr “master” and N. hiril “lady” as derivatives of ᴹ√KHER “rule, govern, possess”; the words ᴹQ. heru/ᴹQ. heri reappeared as well (Ety/KHER). All four of these words reappeared frequently in Tolkien’s later writing, though S. hîr was more typically glossed “lord” (PM/210; SD/129; VT41/9; Let/382; UT/318). The root √KHER itself reappeared in a 1954 letter to Naomi Mitchison with the gloss “possess” (Let/178).
The root ᴹ√SAM “unite, join” was a later addition to The Etymologies of the 1930s with the derivative ᴹQ. samnar “diphthongs” (Ety/SAM). There is also evidence for it in the word ᴹQ. sampane “combination” as in ᴹQ. Lámasampane “Combination of Sounds”, a term used in the first version of the Tengwesta Qenderinwa (TQ1) from the 1930s (PE18/40), and again in the second version (TQ2) from around 1950 (PE18/90). In an isolated note from the late 1930s, Tolkien gave ᴹ√kam “bind, join” as a replacement for √sam along with a new word ᴹQ. okamna “diphthong” (VT44/13), but given the reappearance of Q. sampanë in TQ2 this may have been a transient idea, and in any case Tolkien used the word Q. ohlon for “diphthong” in the 1950s and 60s (VT39/9; VT48/29).
In notes grouped with Definitive Linguistic Notes (DLN) from 1959, Tolkien gave √SAM as the basis for Elvish verbs for “to have”, with Q. samin and S. sevin “✱I have” (PE17/173). Whether this was connected to 1930s ᴹ√SAM “unite, join” is unclear. In notes associated with the 1959-60 essay Ósanwe-kenta, Tolkien gave the root √SAM with the gloss “mind, think, reflect, be aware” (VT41/5), but in later writings he used √SAN for “think, use mind” instead (PE22/158); see that entry for discussion.
Neo-Eldarin: For purposes of Neo-Eldarin, I’d use √SAM = “have”.