You're trying to justify getting a more expensive apartment when you first move down there because you think it will provide a social life. This is one of the things you need to stop doing, justifying overspending.
You need, right now, the cheapest housing that is safe. Don't count on Section 8, I don't know what it's like there, but in my city, the wait list takes YEARS. Go ahead and apply but don't expect it to happen for a very long time.
You can include transportation and utilities in your calculations: a place that is walking distance to work would save you money, being on good public transportation might allow you to give up the car if you have to. Don't choose housing based on what you HOPE to make, but on what you WILL BE MAKING NOW. You can always upgrade your living situation later but right now you need a budget that works with your current income.
Consider looking for cooperative living situations: for instance, an elderly or disabled person who can offer reduced cost housing in exchange for assistance with shopping and personal care. Or look for shared housing with another single mom.
