If there is one thing I learned from the app process, it's that timing is critical.
irmunky: for what it's worth, I would absolutely not wait.
My story is a little different, but basically I had to make the same decision last year. I took the MCAT June 13th and did just "okay" on it. Convinced I could do better, I retook August 22nd and told all my schools to hold my app for consideration until they received my second set of scores. While this prevented me from being "screened out" based on my first score, my 8/22 MCAT was not uploaded by schools until THE FIRST WEEK IN OCTOBER (a week later than it was supposed to be released). Although I initially thought holding my application was a good idea (and it probably did help me avoid the "pre-screening" at some schools), by the time I was actually "complete" at my schools, most of my friends with similar stats and experiences had already booked all of their interviews. With most schools operating on a rolling basis, it is imperative (CANNOT STRESS THIS ENOUGH!) to get those apps in and complete ASAP. Like you two, I kept on top of them and didn't hold on to them for more than a week or two after receiving them, but my MCAT -- or in your case, a letter or two -- will stop them from even looking at your file if you ask them to do that.
On the other hand, if you think those letters are truly going to make your app, maybe you should wait. Honestly, the process is more of a crapshoot than it should be in some respects, and I don't think you will find a "right" answer.
I guess the one piece of advice I have to offer is, if the rest of your app is solid, go ahead and tell the schools you are complete (unless the letters are coming, like, within the week .. don't let them hold you up). One thing I wish I had known going into the process is that schools LOVE to hear from you. They love getting "updates" on what you're doing (well, as long as they are mostly relevant to your app haha), and if you do secure an interview, it's become kind of custom to write a letter of interest or intent (LOI) to push you to the top of the applicant pool. While I don't recommend "stalking" them, if they see your name every once in a while around the office, the committee is much more likely to think of you when a spot opens up. In that way, the "later" LORs could actually work to your advantage, kind of keeping your app "alive" and on the top of the massive stack of other applications.
To sum up, if you're nervous about submitting without, you could always tell them you're complete but drop an e-mail or letter (or both, as I always did, just to make SURE they got it) stating that their office should be receiving LORs from __________ whom _______ (i.e., you have worked with in a clinical setting for x years, etc. -- you don't have to go into great detail, but just so that they want to wait for the letter knowing it will likely be an honest reflection of you) ESPECIALLY if it's someone you shadowed, etc. so they know it's not just a BS letter from your neighbor or something .. yes, people really do that).
Oh, last thing, I promise, ha. You can add LORs through AMCAS for as long as you want. Problem is, most schools download your letters once and only once. So if some will be submitted later, you have two options: 1) submit through AMCAS but make sure you tell the schools that additional letters need to be downloaded, or 2) your letter writers may mail them directly to the school. Talk to your advisor about this second option -- he/she may want you to make sure your writer signs his/her name across the envelope flap, uses a seal, etc. to prove that you waived your right to see the letter.
Anyways, I could (obviously) talk about this for hours .. it's a decision I struggled with when I applied and to this day wonder if things may have been different had I chosen the other option. I definitely get your dilemma. Feel free to PM me, and best of luck with your applications! 🙂