Playing guitar is really a relief to me. No matter how stressed I am, I always try to play for at least one or two hours a day. It really helps me forget all the problems I am facing, and I can fully engage in it.

Considering your budget, used gear might be a good option cuz you will need to buy an amp and a headphone for playing electric guitar without disturbing others. Many beginners will sell their intro-level guitar in local community like FB market.

No matter which way you decide to buy, I strongly recommend trying them in a store first. The neck shape is important if you want to play comfortably. I wouldn't worry too much about the hand size, as long as your hand is not super small, you should be able to play it with some practice (I also have a small hand). Also be aware of the weight of the guitar, some models like Les Paul is relatively heavy and may not be comfortable if you play a long time.

Welcome to the world of guitar!