David Allen would say the full benefits of GTD aren't realized until your mind is completely clear of any open loops and un-captured "stuff", and in general I've found that to be largely true: the difference between using GTD a little--or even 99%--and having everything is discontinuous.
So the goal should be to put everything in your system. However, it is possible to ramp into GTD as you're describing.
I'd start with defining context lists and then add things to them as they occur to you--both new actions, and as they pop into your mind, old ones. Make sure you have a ubiquitous capture system in place, even if it's something as simple as a hipster PDA. The key here is to make sure everything new is captured into your system.
This first step should allow you to prevent things from getting worse by capturing old actions and to clear out of your mental to-do list those things which are naggingly important enough to keep coming up unbidden.
A critical discipline here is to realize that nothing is too small to go on your list. Either the task is so easy (David Allen recommends a threshold of less than 2 minutes) that you should just do it right then--when it occurs to you--or it goes on your list (obviously < 2" tasks that can't be done in your current context need to go on the list as well).
At first this may seem overwhelming because the list will get so large so quickly, but this type of thinking is like not getting tested for a terminal illness because you're worried you might have it. GTD is no more responsible for you having lots to do than a biopsy is responsible for giving someone cancer. Avoidance will not make things better; it will only hide the problem and lead to mental anguish and increased stress.
Now in terms of working through your obviously significant backlog, it's critical to manage this like you would any other task. Break down the project of capturing and processing the rest of your life into manageable chunks and put these tasks on your list. For example, your tasks to process the rest of your life might look something like this (not yet sorted into contexts, so this is an unprocessed list of actions).
process stack of papers on bedside table
read 1/2 of the pile of mail on the dining room table
clear one stack of papers off my desk at work
As you find your motivation to do this lagging, remember that having things in a system is going to save you time because things only get processed once and then you don't have to think about them again until you're ready to do them.
If you find yourself staying motivated and still unable to catch up or stay on top of your responsibilities, then it's likely you are just over-committed. And GTD can only help you solve this problem by identifying it. Re-negotiating your responsibilities falls back on you.