Try any route to get out of a desperate situation, no matter how unlikely it is to succeed.

It is only since the mid-19th century that we have been clutching at straws. Even more recently, the ‘grasp at straws’ version has become commonplace, especially in the USA. Prior to that, desperate people would ‘catch at a straw’. That usage of ‘catch’ was commonly used in medieval England, by which was meant ‘obtain/achieve’; for example, John Wycliffe used it in his 1382 translation of the Bible into English, in 1 Timothy 6:12:

Stryve thou a good strif of feith, catche everlastyng lyf