So it looks like it's wool, which means that you can hand-wash it unless the dye absolutely requires dry cleaning. Wool felts with heat/friction, so you want to wash it the same way you'd wash a nice wool sweater, except in a bigger container. When wool felts it shrinks. The blanket is probably already felted but if it's recommended that you don't wash it then it probably has the potential to shrink more and you don't want that.

The way I'd do it is fill my bathub partially with cool water (doesn't need to be ice-cold or anything and it doesn't need to be a ton of water), add a small amount of shampoo or gentle laundry detergent, and let the blanket soak for a few hours. Then swish it around in the tub without agitating it too roughly and when you're satisfied that the sweat and oils have had a chance to wash off, start rinsing. Drain all the water out, squeeze the blanket out gently and replace with clean water. Do this a few times. (This is why you don't use large amounts of shampoo or detergent, because you'll be rinsing it out forever.) When you think it's rinsed clean, squeeze the water out without wringing.

If at all possible, lay it flat to dry. If not, you can get most of the water out by rolling it up in some towels and squeezing, and then try to support it as much of the weight as possible while it's hanging so it doesn't stretch out. Wool is *heavy* when it's wet.

If it gets stained, spot-treat the stain and dab it out, don't rub.

The good news is that wool is a fantastic material. It's breathable in a way synthetics aren't, so it wicks sweat away and doesn't seem to get nearly as smelly as synthetics do. It still feels warm even when it's wet. It's biodegradable.

The bad news is that it is scratchy. Virgin wool is better than some heavy-duty wool I've knitted with, which has the approximate texture of a heavy-duty dish scrubber. And it will get softer over time... but it's not going to be as soft as acrylic or microfiber. If it continues to bother you, use it with a top sheet or duvet cover and then you can have the benefits of wool without the itchiness.