Mushrooms are just the fruiting bodies of various fungal organisms that are present in any kind of decomposing organic matter. They tend to be temporary and typically harmless and will pretty much vanish from the scene once the weather warms and dries.

If livestock manure is not composted to a sufficiently hot degree, it can sprout weeds from the seeds in the fodder eaten by the livestock in question yet not fully digested. Typically dairy manure has less of a problem with this than horse manure due to the complexities of the cows' digestive system. One most often encounters grain-type weeds as the result of not fully composted cow and horse manure. It is just as likely that the weeds you describe have germinated from seeds blown in by the wind and not present in the manure itself - manure is a pretty fertile medium for the growth of any type of seed/plant, provided it is not still "ripe" or fresh.

I do feel for you.It happened to me before and I had lots of work to keep removing the sprouted greens ( I had no mushrooms though) IF this is of any help the greens you remove become a green manure also good for your garden.With time you will win this battle for sure.Good luck

Thanks much for those replies. I'd say it makes me feel better to know I'm not the only one that this has happened to, but it sure makes me nervous about ordering another load in the future!

My guess the weed is Purslane. If it is, you are going to have a hard time getting rid of it. A bark or woodchip mulch over the top of it would be one way to help control it. Do not let it go to seed! Pulling it and then leaving it on the ground, upside down or not, won't work. It will sprout roots and start growing right away, even in the sun.
The weed is common in barnyards and gardens in Central and Eastern Washington. I feel for you also.

Here is a link that might be useful: Purslane