Hi, and welcome theo the Commercial Section!
Please post some images that you have already made of of your clients. This will give me some idea of where you are at and what you are upto. This will help me advice you on setting up a simple but effective system for consistent results.
Flat ring lighting is not the best solution. There is nothing wrong with "shadows" providing they are placed properly. A good lighing system should render texture and color in the hair and also show how your styling flatters the facial structure. Lighting the hair itself is an important factor in bringing out highlights and brilliance and color.
Some of the principles of portrait photography factor in as well, so lens choice, focal length and working distance is important to avoid distortion. Camera angle is very crucial in hairstyling illustration.
Tell me where you will be shooting- at your salon or location or at a locations where portabel gear is needed? If it is at your premises, a permanent and easy to use set up can be placed in a corner of the room where the client can easily take a seat and pose for a few minutes.
If you have not as yet purchased a new camera, perhaps hold off until you gather all the information.
If you shoot correctly, frame carefully and light and expose accurately, you should not require massive editing. Believe it or not, many photographers tend to shoot sloppily and edit their heads off to correct issue that coud have been avoided by careful shooting.
Also- tell me what end use of the image will be- display prints in your salon, a portfolio, Internet ads, screen images- a slideshow- wahtever? This will help in creating and designing a system.
The best tip I can offer you now is that any decent DSLR or mirrorless camera should do the job with the correct lens. The technique is more importat than the gear and you need to invest in a simple but reliable lighting system. Most importantly, don't forget to post some images- if there are any problems or issues they can be worked out!
I am shooting all day today- so if you post something soon, I will get back to you by Tuesday or Wednesday at the latest.
Sound like a cool project! Ed