I related to your post, and sug', I'm sending you so many hugs.
Look, when my daughter was around the same age, I felt pretty isolated and fragmented, in no small part because I was. I also live a ways out of town and I gotta tell you, the winters here are dreary.
I'm a person who does best when I'm a whole woman. When my daughter was very young, I had some trouble discovering my identity, as a mother AND all of the other stuff I had already been. It helps me to think about myself as a watch. I cannot extract the woman from the mother, and if I don't tend to my whole self I feel pretty miserable.
I don't know about you, but I like being around people. I need alone time too, but social companionship is a very important process in my life. I have learnt for from my relationships (and the conversations I get to have in them) than I have from any other single source. Granted, it can feel overwhelming at times, and I've also got to take time for myself, just to decompress.
But anyway. At some point I figured out that I wanted to model to my children a fulfilled parent. So I started having more fun in life. I started reading a lot, just between things or when the kids (by then my niece had started living with us) were napping. I called up friends to chat while I emptied the dishwasher, or took my children for adventures that I was excited about (why NOT play dragons in the woods, or just lay in the grass staring at a bug? Why NOT vacuum in heals just to see if I can?
Yes my house is a mess sometimes (okay, most of the time), and yes I have off days. I don't play with my kids ALL the time and I take space from them when I can (I set up childcare trades with friends). Once or twice a day, I play a show for them or let them play starfall.com so that I can daydream on mamapedia.
In short, I forgive myself for being human and I do stuff that keeps me excited about life. Because I want my kids to be excited about life. I want them to see me as curious and engaged. I want them to see me with a close circle of friends and a healthy relationship with my self and others. Because, more than anything, I want that for them. I want them to feel fulfilled and I figure the best way to help them to do so, is to learn how to do it myself.
I don't think you sound like a bad mom. I think you sound like a mama who feels scared and guilty, and, well, I don't think many of us haven't felt that at times. At least, I've felt that way. A lot.
Trust yourself. Stay in the present moment as much as possible (and let yourself off the hook when it's not perfect), and do things for yourself. Being a parent is tough work. You deserve to feel good about yourself. Independent playtime is a great skill for children to learn, in my opinion.