Try using the "hourly rate" in the teacher/tutor search. This will narrow down the results to just what you can afford. Some teachers/tutors also offer 30minute classes so if you can't afford the hour you could do 30minutes. I hope this helps :)

Yes new teachers generally start with lower prices. e.g. I am a new teacher with plenty of experience in classroom and online teaching with other companies and I am available on italki for $11 per hour! I too will raise my prices at some point in the future when I get more students.

Generally speaking you will get what you pay for. If you really want to pay as little as possible try to find the newest tutors. People will start with lower prices until they get some students. In this way it's hard to confirm if they are good or not but I have found some very wonderful teachers this way who even extended me a discount to keep learning with them when they raised their prices for new students.