I just found a great app. So great, in fact, that I actually paid for it. First time for everything, I guess.
It's called Waterlogue and it's available on the App Store for $2.99.
After some trial and error, I found that the best candidates for these photos are closeups without a lot of background detail. You can choose to add a border if you like.
There's a chance that I may be become obsessed with this app and every photo I post to Facebook or Instagram will be a watercolor.
Love this!