「乾」 (Baxter-Sagart OC: /*[k]ˤar/) is the original character meaning dry. Using 「幹」 for dry was sometimes found in history, but in Modern Chinese this should be considered a spelling mistake.
「幹」 is a corruption of 「榦」. 「榦」 is comprised of semantic 「木」 and phonetic 「倝」, indicating the meaning tree trunk, extended to mean main body > main organisation, cadre, framework > to conduct, to do. 「木」 was sometimes corrupted into 「干」 to give an additional sound hint, producing 「幹」, and this is the variant that the Taiwan and Hong Kong standards use.
「干」 depicts a two-pronged weapon, but this meaning is no longer used. Most other meanings you find with this character are phonetic loans; it is now most commonly used for the Heavenly Stems (天干地支), unless you're learning Simplified Chinese.
「乾」 is sometimes pronounced qián (OC: /*[ɡ](r)ar/), meaning sky, heaven. This is a phonetic loan.