I recently ran my circa-1981 brown tweed ones through the washing machine on a hot cycle with laundry soap and a splash of Lestoil to help with the diesel smell. The threadbare spots didn't do well, but the rest is OK and it really killed the smell. If there is a tag somewhere on it or you can verify they are acrylic, they will not shrink, nor should they bleed. Search the internet, there was a chart from Sunbrella about what to use on various stains. Washing them will take out any stain resistant or waterproofing treatments (Scotchguard or 303) and those will have to be re-applied (which is possibly more important for exterior covers).
Smells also collect in the cushions and I washed mine by hand in the bathtub with a similar hot water solution to try and kill the smell. That was moderately successful, not perfect, but bearable to my sensitive nose. Wet cushions are very heavy and took days sitting outside on the back deck in the sun to dry. If needed, there is a process for getting the cushions back in the covers using plastic cushion silk and a vacuum cleaner to compress the cushions so you can get them back in. I think it was on the Sailrite web site. They sell the silk also.