Generally the feed store and/or Co-ops do test hay BUT their labs generally test for cattle not horses. You would have to ask if they know for sure hay can be tested for the things you need to know for a horse.
I bought a core sampler and send samples to Equi-analytical in New York.
You can pick n pay the type of testing you want done, their turn around is fast, and they either e-mail the results or send them thru the mail. Equi-Analytical Laboratories - Profiling Feed for Better Nutrition
It would be great if your local Ag Center, Co-op, or one of the feed stores had a core sampler they loan out so you don't have to buy one.
This Triple Crown link provides a general idea of the sugar/starch content in some forages. Horse Nutrition: Triple Crown's Carb & Horse Nutrition Recommendations
Not trying to scare the OP but Drafts are on the predisposed list for metabolic issues, so please take your horse's need to lose weight seriously.
The link above also provides sugar/starch content for various types of grains. Please note that oats and corn are the worst things over weight and metabolic horses can have.
That means no bagged feed that has oats and corn in it, unless it specifically says "distillers grain". Distillers Grains means the sugar and starch have been sucked out (for lack of a better word) for use in making whiskey
Hope this helps