Hmmm. It sounds like the front of the lines' fittings aren't seating fully against the base of the radiator's tranny line bungs. Those lines shouldn't be able to be jiggled about in there like that - they're the same as the brake lines coming off the side of your master cylinder, once they're seated they should be firmly affixed in place and not able to be wiggled around.
Is this the same set of tranny lines and radiator that you'd had bolted together before having the tranny serviced? Could be that there's a difference in the fittings you're using. Like, the lines may be a flare fitting of one kind, but the hole(s) in the radiator aren't the same. Hard to describe. Look through a bunch of brake line and pipe fittings in a Dorman drawer at a local Autozone or Checker or whatever, and you'll see what I mean. Some have a flat front edge, others have a beveled front edge or inside lip, and so one will not thread into another and seat correctly.
If your tranny cooler lines and/or the radiator have changed, then it's likely that you just need to use a couple of brass brake line or transmission cooler adapters (don't forget the Teflon). Again, try looking through the Dorman drawers at your local parts store(s), and also look over by where they sell their aftermarket tranny coolers, as they'll usually have a packaged assortment of adapters available there - I had to use a pair of 'em when I added a stacked-plate B&M tranny cooler to a 94 Cougar I used to have.
Don't try to RTV the fittings, as it won't prevent leakage in this instance, and you'll just gum things up.