What I don't get is why joining the NSA isn't something like getting a really well paid job combined with being in the military.
Pay them really well, so well they would have to think 3 times about not joining. Like 4x a similar pay rate that you'd find in a top-tier city for an equivalent job. Make working conditions really nice -- free high-end restaurant quality dining on premises with a room service option for people who wanted to work through a meal hour, super nice office spaces, the whole experience more "hotel" than "government office".
But then also kick in the military part -- you join up for a minimum 5 year commitment, you live on campus, your travel off-campus is limited and controlled and there's the understanding that you ARE being watched closely, but do it in an unobtrusive manner, not in a police state manner. But make the housing and lifestyle options more like a country club kind of atmosphere, single family houses, lots of recreational options, private schools for the kids, and lots of activities for spouses and kids, too. Make them stay but make staying so easy they want to stay.
Sure, the whole thing would be expensive, but you'd have a much better chance of containing your secrets. And chances are, buying their loyalty would go a long way to helping and keep the security more velvet glove than rubber glove.
The current thing with all the contractors is a mess and it's a miracle that actual government employees have any loyalty at all.