In Tolkien’s later writings, the root for Elvish “sea” words seems to be √GAY(AR) or √AY(AR). The first hints of this root may be words from the Gnomish Lexicon of the 1910s: G. ail/ᴱQ. ailo “lake, pool” and G. ailion/ᴱQ. ailin “lake” (GL/17), perhaps indicating a (hypothetical) early root ✱ᴱ√AYA(LA) “lake”. The last of these Early Qenya words also appeared in the contemporaneous Qenya Lexicon, but Tolkien gave its root only as “?” (QL/29). This early root might also be the basis for 1920s ᴱQ. ailin “shore” which appeared in the Oilima Markirya poem (MC/213).
The word ᴹQ. ailin “pool, lake” reappeared in The Etymologies of the 1930s derived from the root ᴹ√AY, with its Noldorin form N. oel reflecting Noldorin rather than Gnomish phonology (Ety/AY). This root had an extended form ᴹ√AYAR “sea” from which Tolkien derived ᴹQ. ear and N. oer of the same meaning; among other things this extended root was the basis for a new etymology for the name ᴹQ. Earendil as “Friend of the Sea” used from the 1940s and forward (SD/241, 305); in earlier writings ᴱQ. Earendel was connected to ᴱQ. earen “(young) eagle” (QL/34).
The root √AY(AR) “sea” continued to appear in Tolkien’s later writings, for example in the second version of Tengwesta Qenderinwa from around 1950 (TQ2: PE18/97). However, Tolkien also considered alternate derivations of “sea” words from a new root √GAY(AR), for example changing √AY(AR) >> √GAY(AR) in etymological notes written between the 1st and 2nd editions of The Lord of the Rings (PE17/27). As opposed to √AY(AR) which meant simply “sea”, Tolkien also glossed √GAY(AR) as “astound, make aghast” (WJ/400) or “awe, dread” (PM/363) and it was the basis for other words such as S. gaer “awful, fearful” (WJ/400) as well as S. goe “terror, great fear”, Q. aica “fell, terrible, dire” and Q. aira “holy, sanctified” (PM/363). In this sense √GAYA may have been a replacement or a variant of √AYA(N) “blessed”.
Tolkien seems to have been unable to make up his mind between these two derivations, as reflected in an ongoing vacillation between S. gaear (PE17/27; PM/363; WJ/400) and aear (Let/386; RGEO/65) as the Sindarin word for “sea”. This word appeared in The Lord of the Rings in the phrase nef aear, sí nef aearon “here beyond the Sea, beyond the wide and sundering Sea”, but that does not resolve the question as the word in this phrase seems to be lenited, and hence would lose its initial g (if any). This vacillation continued late into Tolkien’s life: in the published corpus Tolkien derived “sea” words from √AYAR “sea” in a letter from 1967 (Let/386) and √GAYA “awe, dread” in The Shibboleth of Fëanor written in 1968 (PE/363).
Neo-Eldarin: For purposes of Neo-Eldarin writing, I think it is best to use √GAY(AR) as the form for this root, though admittedly this does create problems for the etymologies of Q. ailin and S. ael “lake” (these would need to lean more heavily on their second ancient element: √LIN “pool”). It is not clear whether Tolkien always intended √GAY(AR) “awe, dread” and √AYA(N) “blessed” to coexist, though he did at least some of the time (for example on PE17/149). For purposes of Neo-Eldarin writing, I think it is best to assume they did coexist, with √AYA serving as the basis for “holy” words like Q. aira/S. aer “holy” and Q. Ainu, whereas √GAYA could serve as the basis for “awe, dread” words like Q. aica “fell, terrible, dire” and S. goe “terror, great fear”. Q. ëar/S. gaear “sea” would fall into the second group as a thing inspiring awe.