Oh FFS - where does all the 'if he wanted to get married, he'd ask' come from?
This is not 1920. it is 2015. You are 2 grown adult people. You love each other - yes? You have discussed, and are close to, moving in together - yes? So presumably you have talked about your relationship, where it is going, your expectations for the future, your joint plans for work, career, retirement, the management of family and elderly parents - yes?
[or maybe you've not talked about any of this, in which case - that's a good place to start]
So why not talk about marriage and your expectations and your plans, and then agree what you want to happen....
And I don't understand why when he asked you to marry him, you think that is not a 'proper' proposal. He asked you to marry him. That is a proposal (unless he was drunk, incapable etc). He'd been through something traumatic, and the most important thing to him was asking you to marry him... so what did you say? Did you reply? Does it not count as a 'proper' proposal unless he is on one knee and you are in Italy at sunset, and he has the perfect diamond in his pocket?
You are about to move in together. That is a major commitment for him, for you, for your dc. Never mind the niceties of who asks who etc. Just work out (together) what you commitment is, and then go for it.
On a personal basis, I was in a similar position to you (I had dc, he didn't). We were long term, committed, in love, best friends. I asked twice about marriage - my first proposal was the 'theory' "If we are going to do this [serious living together] then I want to know that we will get married" and then my second was "OK, now is the time - will we get married this year?" If he had said no, or if he had prevaricated, or delayed, or said he wanted to ask me (when the [mythical] time was right) - there would have been no way in which he would have been moving into my house, with my kids. Harsh, maybe? but love and marriage and commitment is about more than a romantic ideal - it is also about a very practical set of family arrangements.
OK, so that all sounds unromantic and logical and mundane? But my dh could tell you the exact spot in France (in the sunshine,, by a river) where the first proposal happened, and our second engagement was in idyllic frosty conditions which are imprinted on our memories - more deeply romantic to us than a 'down on one knee' proposal.