Call a carpet cleaning company and have them come into the house to clean them using the carpet cleaning stuff.

The vendors that come to the house use a steam extraction method.

There must be a good old-fashioned dry cleaning method that was used for traditional furniture at one time, before the steam franchises became ubiquitous and furniture with no zippers became the norm.

Anybody know how I could find something like this?

Doing the cleaning with steam while the covers are on the cushions does prevent shrinkage, I suppose. But it presses the dirt into the cushion, and I'd rather not have a big truck and crew come to the house for my little loveseat.

You may be thinking "Why doesn't she know all about this stuff?". I don't have carpeting -- I send the orientals out for water cleaning -- and I used to vacuum and spot clean my cotton tapestry sofa myself because the pattern hid stains.

This fabric is poly.