I don't know much about shrimp, but I do know that shrimp are sensitive to water. You'd also need a few shrimp (depending on size) which would produce waste as well. I'd honestly suggest trying to find a bristlenose pleco, if you really want to have some help with getting rid of the algae.
OP has only a 15 gallon tank and is right that it is too small for a bristlenose, I think in general, but even more so if the 6 corydoras are not dwarf species (habrosus, hastatus or pygmaeus). Amano shrimp are quite hardy and are very good algae eaters. Shrimp make very negligible waste compared to any fish, but especially when compared to plecos, which are iconic waste-producers, will hardly expand the bioload at all. OP could safely have quite a few amano shrimp (at least 5 would be fine, probably more would be OK too but might start looking crowded), and they would help a lot with the algae. Petco often sells them cheaply as "Japanese algae-eating shrimp."
Like others have said, though, if you localize the problem, which is likely light and just an immature tank, the algae will go away naturally. Too much light is more likely the issue. What is your current photoperiod?