If you are interested in the possible implications of making such a request to your advisor or accepting such a position, please check the linked question and all the related answers. I also gave an account me asking my advisor about it in an answer to that question, but the focus was a bit different so I'll summarise the relevant parts here.
Staying at the same lab following PhD graduation for a short position (~6months) can be beneficial to the student to ease the transition period. Longer positions are mostly beneficial to the lab only, but not to the academic career of the (now former) students.
Therefore, I think one of the possible reasons you have not received an offer from your current adviser could be that they are not aware you might need one (it could be, in a very minor way, a compliment).
I was in exactly your situation a month before my viva: looking through some job adds, not really finding any good fits, and spending 10+ hours a day on my viva preparation. So I had decided to ask my adviser for a short postdoc position to help me out of the situation. I was very direct. I'd invited him to go for our lunch break together (basically, just sit together during the lunch hour in the University canteen, where we would both typically take our lunch anyway). Some small-talk appropriate for the culture, and then I flat-out asked: "I know I should've, but due to my intensive viva preparations, I did not yet sort out a (postdoc) position for after my graduation. Could you help me out somehow?"
As a result, he'd offered me a 4-month postdoc, on the (legally mandated) minimal postdoc salary. The funding for it came from the discretionary lab budget, not connected to any project. I have seen a lot of (established) labs in Europe will have that type of budget.
While I can not help you cope with your feelings of awkwardness, I strongly believe nobody should feel ashamed to ask for help when they need it. The worst that can happen is being told no, and even then you are likely to end with more information or at least advice then if not asking at all. Especially in this situation, as your are asking for advice on your academic career from the person who is supposed to provide such advice in the first place.