Out of desparation, the act of reaching or stretching for a solution, no matter how irrational or inconsequential. Literally - a drowning person who, unable to find any substantial floatation immediately at had, attempts to save himself from certain death by grabbing hold of a few stray pieces of straw floating on the water within arm's length.
The Senator from Florida, caught red-handed sending lewd and lascivious text messages to underage male pages, was clutching at straws for an alibi and escape. Alcoholism? Resignation?
This seems to be a mixture of "led down the garden path" (meaning misled) and "sold up the river" (meaning "handed over to law enforcement", slang derived from the fact that Sing-Sing state prison is "up the (Hudson?) river" from New York City).
The enthusiastic young first-time buyer was led down the garden path by the unscrupulous used-car salesman.
The entire gang that pulled off the bank heist was sold up the river by the getaway driver after he was picked up and interrogated the next day.