[verb て-form] + しょうがない / しかたがない [i-adj 〜い → 〜くて] + しょうがない / しかたがない [na-adj] + で + しょうがない / しかたがない
てしょうがない and てしかたがない say a feeling is so strong you can't do anything about it: "unbearably ~", "I can't stop ~", "dying to ~". They follow the て-form: an い-adjective takes くて, a な-adjective で, and a verb of feeling its own て-form.
The last one is the commonest verb case: たい changes to たくて like any い-adjective.
It takes feelings and sensations that come over you – worry, loneliness, sleepiness, pain, wanting something – and not a deliberate action. The feeling is the speaker's own; for someone else's, add ようだ or らしい.
The two mean the same. しょうがない, a contraction of しようがない, is what you say in conversation. しかたがない, often written しかたがない仕方がない, is a little more neutral and is the one for writing; in speech it drops が, しかたない. The polite forms are しょうがありません and しかたがありません.
With も the meaning changes. てもしょうがない is "there's no point ~ing": the action is useless, not the feeling strong.