Hi T.,
I was a single mom in a situation very much like yours from the time my son was born until he was six (when I met the love of my life and we got married). Reading your post, I remember feeling exactly the way you are now. With your son just 5 weeks old, you are in what I thought was the hardest part of being a new mom. But, as your son grows and you get more used to being a mom, it does get easier, literally each month. Also, in addition to being in a tough spot, you might have a little post partum depression, so if you continue to feel down and overwhelmed, see your doctor.
In our case, I can't say that there was a lot of balance between being a mom and having time for myself for a long time. Like yours, my parents were helpful, but I still had all the responsibility. Finding daycare that I trusted helped me actually feel good about going back to work because I knew he would be happy and entertained there, and going back to work was good for me because it reminded me that I was still my old, capable professional self. As far as taking time for myself, I had a job where I could schedule a haircut or nail appointment during my lunch now and then, and that helped. But that was pretty much it.
As for keeping your stress from affecting your son, that is tricky because you are human and at points, he is just going to see you be human. I can remember breaking down and crying while folding laundry in front of my son when he was about two because it all just seemed like too much. I think as long as your stress isn't causing anger or you to mistreat him, things like that are going to happen. If you do feel more negative aspects of stress, try to find help through a professional counselor.
Finally, one thing that really helped me was finding a coworker that was also a single parent. After we got the kids down to sleep, we'd call each other and talk and laugh and have a glass of wine over the phone with each other. So even though we were not going out, it was a time to just be ourselves for a while. Finding a friend or two like this could help a lot.
Finally, it helped to just focus on my son and have what used to be "my" time become "our" time. On Saturday mornings, instead of going to the gym like I used to do, I'd put him in his stroller and go for a walk with him. One benefit of being a single parent that dosen't really get talked about is the bond that you can create between yourself and your child. Even though we have a dad in the picture now, my son and I are so, so close and I really value that.
Good luck and just take it one day at a time. Honestly, sometimes it will be just one hour at a time. Just remind yourself that you are great and strong and can do this.