The noun G. saith “hunger” appeared in the Gnomish Lexicon of the 1910s (GL/66), like based on the early root ᴱ√SAẎA “hunger” which was mostly used in combination with -kǝ (QL/82). In Early Noldorin Word-lists the word appeared as sais, revised to ᴱN. hais (PE13/147, 153) to reflect the fact that initial s generally became h in the Noldorin of the 1920s (HIPTN/§4.1.1). In this 1920s document its Qenya equivalent was ᴱQ. saiste indicating a primitive form of ✱saistē.
Neo-Sindarin: For purposes of Neo-Sindarin, I use ᴺS. saeg for “hunger” based on the Neo-Root ᴺ√SAYAK, a neologism coined by Fiona Jallings based on ᴱQ. saike “hunger” (FJNS/352). An argument can also be made for ᴺS. saeth as a more direct update of G. saith (perhaps < sayak-tē as suggested by Elaran in 2018), but I prefer saeg since its etymology is more transparent.
The adjective G. saig “hungry” appeared in the Gnomish Lexicon of the 1910s (GL/66), clearly based on the early root ᴱ√SAẎA “hunger” which was mostly used in combination with -kǝ (QL/82). This word reappeared in Early Noldorin Word-lists, but was revised to ᴱN. haib (PE13/146, 153). Here the final b probably represents a shift of the ancient form to ✱saik-wā, while the initial h reflects the tendency of initial s to become h in the Noldorin of the 1920s (HIPTN/§4.1.1).
Neo-Sindarin: For purposes of Neo-Sindarin, I use ᴺS. saeb “hungry” as an update of the 1920s form to better fit Sindarin phonology. Fiona Jallings first coined this neologism based on ᴱQ. saiqa (FJNS/352).