I am a civilian employee of the military and  I have the same concerns as you.  I was really uncomfortable giving out my work info.  I ended up seeing a very well reviewed and long time TER provider for my first time.  She white-listed me and has been very helpful with references.  That was several months ago, but I still occasionally get requests for full name and work #.  I don't like it but if you want the provider to feel comfortable meeting you it sometimes is required - even after getting references.

Here's one good way:  Go to date-check.com, browse providers in your area, and call or email one or more that appeal to you (checking their TER reviews first!). You will probably find one who will see you w/o references if you are very honest and straightforward. Once you've seen a date check member provider, ask her for a date check referral. Then go back to date-check, look up her page, and click the 'we've met' button. Once she has verified you, you will be able to join date-check without employment information and will have 1 referral. This will open a lot of doors....

If you have already seen 3 providers multiple times, why don't you ask one/all of them to vouch for you?

When you are stateside, isn't there a phone number for your unit that someone wanting to screen you can call to ask for you and leave a message?  Assume for a moment that you don't have a cell phone, and someone asks you for a daytime contact number, what number do you give then?