Shih tzus. Affectionate, ok by themselves for about 5 hours or so, easy for older relatives to look after (not needing energetic runs). Whoever picks your dc up from school could collect the dog as well, or just have a dog walker take your dog out for company and exercise halfway through the day. Adorable cuteness is an added bonus.
Is your son 4? Most rescues won't allow you to rehome a dog with a child under 8. Not dog willbe happy on its own all day and I would be concerned about taking in a dog with a "history" that you might not be fully aware of with a small child in the house.
One possible way of exploring your suitability, desire and commitment to a dog, is to help someone else with their's.
I used to have a neighbour whose situation was very similar to yours, OP. She was a lone parents, working, two DCs in her case, limited family help, and she thought she might like a dog, but had no real experience.
She started by helping out another set of neighbours, a retired couple, with their dog. She very regularly shared dog walking duties with them, had it with her if they went away for the day and, over time, started to look after it for longer periods, such as family visits, holidays and also when the husband of the couple was ill, in and out of hospital. In this case it worked out well. Her home became the dog's 'second home' and, in turn, the dog's owners taught her a lot about keeping a dog.
It was a good way to learn and I think she got a dog of her own a few years later (if it hadn't worked out, I can imagine it would also be an equally good way of finding out that dog ownership wasn't the right thing).
I would second that idea, it's a great way of seeing what dog ownership is all about. It's great having a dog (I've looked after my brother's dog and fostered for a couple of months but hesitate about getting my own) but it is restricting in some ways, you worry about them as if it's another child. You have to make arrangements if you're going to be out of the house for a long time.
As a single parent, I would imagine that you're juggling lots of things already - you may prefer part-ownership to full-time. But, on the other hand, if you do walk your neighbour's dog for a while, perhaps you could talk to them about helping you out at times you might need it (although if they work full-time might be difficult).
An older dog, well vetted by a rescue does sound a much better solution than a puppy which is really hard work. Your kids will naturally lose interest once the dog's been at home for a few months (however much they've mad about it at the beginning). and the responsibility will be all yours.