Hello and welcome to car-seat.org!
Your sense that your 3 year old is not safe in a high back booster at this point is spot-on. Not only are 3 year olds not mature enough to sit properly in a booster (as you have noticed!), but their skeletons are not developed to the point where the seatbelt's 3 points of contact (shoulder and 2 hips) provide enough protection. A 3 year old needs to be in a harnessed seat in order to be protected in the car. A high back booster is a safe choice for a child who is at least 4 years old (for skeletal maturity) AND is mature enough to sit still in the booster (which usually happens around age 5-7).
So, instead of a booster, you should be looking for a harnessed seat, either a convertible (one that can go forward or rear facing), or a combination seat. Since you won't be using the rear-facing capabilities of the seat, a convertible is probably not the best choice, unless you DO want to have your child rear facing, or if you want to use the seat for another child in a rear facing position (maybe a niece/nephew, babysitting charge, or future child).
Assuming that you do want a combination seat, there are lots of good options, with different features. Generally speaking, spending more money will get you a seat with nicer "finishes," that may last longer, and/or be easier to use properly. A good starting point for your shopping is the
list of recommended seats on carseatblog.com.
Since your child is not adequately protected by a booster, I would try to make a quick decision and purchase, and limit car trips as much as possible until the new seat is ready for use.
Amazon usually has good prices, and quick shipping (paying for 1 or 2 day shipping is probably a good decision if you can't limit travel for very long).
Please post back if you have questions about specific seats!