Hi Lulu,
my mum died six months to the day before your mum, so it is a little over a year now since I lost her. I am so sorry to hear of your other troubles. Looks like you have had a lot to cope with in a short time. Six months in, I can say I was in a complete state of meltdown and stayed that way for a week or two.
Having got through this first year now, I can tell you it does get a bit better, although it doesn't really happen on its own. I think you have to make an effort to find a way to help yourself. For me, I took up a new interest and joined a class, so that I would have to get out of the house and amongst other people to take me out of myself and help me think about something else, if only for a few hours a week and meet some new people. I can say that that did really help me and still does.
I have found it healthy to take a step back from my grief for a while now and then and concentrate on other things, and although the waves of grief do still come over me at times, I find I am coping with them a little better now, largely because I have this other interest in my life to help me keep engaging with life outside of my own personal situation.
With your new puppy, perhaps you could join a puppy training group to help you do something similar, or find something else to do that you might enjoy.
Six months isn't very long in terms of grief, so I am afraid you probably still have a long road ahead of you, but it sounds like you did all you could for your mum in her final illness and so have nothing to reproach yourself for. I think we all take the presence of others for granted at times, especially when we are young and may not yet have experienced loss before. My advice? Make sure you eat and drink, whether you feel like it or not, engage with life and other people in some way, if only for a few hours a week, be patient with yourself and you will gradually come to realise that this is the new normal and accept that you will need to absorb what has happened and slowly, you will learn to live with it. Accept also that it is not something we ever really get over, but know also that things do very slowly get better, even if you still have bad days, there will also be better ones. This is a very slow and difficult process, but take it a day at a time. You will get there.
Thinking of you and sending hugs!