I’m going to assume you meant your skill level is close to @Hackasaurus Rex, since the OP is a 5 handicap.
Anyway, it really depends on what clubs you currently play and how old they are, as well as your skill level. What exactly is it about them that you (and others) feel is holding you back? If you feel like you should get an upgrade as a high handicap player, most people would point you towards previous model year clubs. They will perform similarly to the latest releases, but they’re cheaper. Depending on your skill level, you might not need a tour-style fitting, as a lot of that has to do with finding the exact combination of shaft and heads to fine tune performance. A properly fitted club can only do so much if your ballstriking isn’t very consistent.
Question for ya: my skill level is similar to the OP here in that lessons are going to help me more than new clubs (which I am doing). However, I've also been told by my teacher and from some very experienced golfing friends that I would benefit from some new clubs, as my current are hand-me-downs, and while I did get them sized to me, are not the type that they would pick. They also mentioned that it's possible the re-sizing multiple times has not worked to my benefit.
So, how do I go about picking clubs if a fitting as described by the OP could occur? My friends have actually recommended club champion and I was considering it, but was worried I would have an experience like the OP did if I came in and was swinging poorly.
I'm by no means an expert compared to the people here who have already responded, but figured one more perspective may not hurt.
When I'm in a routine where I can practice every day, I have found that more frequent lessons (roughly every other week) are helpful, mainly to keep me on track (receive feedback). One of the difficulties I have is getting meaningful feedback; obviously we know when we hit it poorly, but I often don't know what I've done wrong. So having a professional take a look and point it out has been helpful.
That being said, I may be an uncommon scenario; I have reinforced a swing with some poor characteristics that have been difficult to shake. Sometimes it takes a teacher showing me multiple times to change something before it sticks. I took two weeks practicing my take-away only to have the teacher show me (for the third time) how I was practicing it wrong the whole time.