DEX 3 is RED (not dirty) so needs to be changed anyway.
Do some search as there are several threads on draining/flushing trans
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Some people put ATF in engine to clean sludge. (Do Not drive it, just idle twenty minutes or so - engine off, Drain immediately, new oil,filter.)
(More better is to simply change oil/filter at say 1000 miles max a few times- this cleans slowly so it doesn't clog something. Avoid city running, much as you can, and try for several 30+ minute runs highway. Engines get dirty from running too long on oil changes, and from not being run long enough at full operating temp mostly. The 'Bottom' oil/crankshaft area takes over 20 minutes running just to get hot and needs a while to cook off the contaminates)( If mostly short trips THAT IS Severe Service)
I would run less than 500 miles and change oil/filter again just to be sure to clear out atf and any crud that flushed loose. And check often, watching for dirty oil for a change or two.
(Stuff happens - Just learn and go on)
Good Luck,
Orange