A six pack of Woolite sheets (that'll last you for 36 washes) and a set of mesh laundry bags (that you can use, like, forever) will cost you a total of $18.77, which clocks in at less than what you'd pay to dry clean two coats. Probably. (Sorry, I live in NYC where everything is grossly expensive.)
Even if you no longer need to wear a winter coat in April, (lucky you!), you should wash it before you put it in storage. Actually, you should've already washed your coat at least once this season. Your outerwear should be clean, period. But also, removing dirt and built-up grime can lengthen the life of your garment.
First, read the manufacturer label! You'll find that most down/puffy coats can be machine washed. There are some, like wool coats, that you'll still have to treat with some dry cleaning (with Woolite) sheets. Remove fur collars, hoods, or any other detachable pieces and, if they can be washed, place them in a mesh laundry bag before throwing them in the wash, too. After you wash your outerwear, (per the tag's instructions), dry according to those instructions, too. Don't freak out when you see your puffer coat reduced to a wet, crumpled shell of what it once was. It'll be fine once it dries.
Get them from Amazon: set of five mesh laundry bags for $9.89, and pack of six Woolite sheets for $8.88