Will I be able to make this work?

I’m moving to the South Alabama and I’ve been looking at places for me and my puppy to stay at. Most of the places are a little over $500 and my parents are saying that I won’t be able to afford it. I won’t be moving until January but I’ve been saving $600 to $700 every month of my checks so by then I should have over $4000 which doesn’t include my birthday money and Christmas money I’ll be getting. I also have almost everything I need and my dog won’t need to see the vet again until next year. I’ll also be working possibly full time depending on my class schedule. And I will be getting a meal plan from the University I’ll be attending to help me save money on groceries. Would I still be able to afford these places if I budget and save? Or am I in over my head?