I hate to be bearer of bad news
but really the walls should have been cleaned before spackle. You want the spackle to attach to the wall not the dust on the walls. Now that you finished the hard work, you can try priming and see what happens just use a good primer, or clean them and go back again to touch up the spackle areas. But you are correct that the TSP will clean out some of the spackle. To me at this point it would depend on how dirty the walls were, did people smoke in that room, was there grease
I would probably clean them if it was dirty then spackle again, grab a beer in frustration wait for the walls to dry then spackle, prime and paint. Or again if they are not dirty just use a good primer and finish the project, most people do not clean walls before priming unless there is a reason.
Sorry, good luck!
Ex Realtor, house flipper, remodelor and now building a vintage truck, making friends and fixing my house.
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