I don't know much about Tushingham Thunderbird sails, but I glanced at the internet and from what I see, they appear to be good freeride sails. Finding a used one should be your goal. At your stage of learning, I think the 6.5 should be your next purchase, but you might also consider a 6.8 or 7.0 given your weigh. You will need a 460 cm mast for the 6.5, so I hope you have one. If not, go for a used 460 carbon mast (40-80% carbon). No reason you can't perfect your tacks and jibes with the 5.5 sail. When you go out, before crusing, do 10 tacks and 10 jibes first with only 10 seconds in between each one. The more you do, the quicker you learn.
If you keep at the windsurfing, your next step after the 6.5 may be to go up to a 7.5 sail, hopefully one that would fit on a 460 mast. You will also be looking for a smaller board sometime soon, one without a dagger board and about 120 to 140 liters. I wouldn't go for the board until you can consistently make tacks and jibes (90% of the time) and are comfortable in a harness and the foot straps.
The harness and foot straps will be needed when you start planing (dagger board up), but it will take at least 15-17 knots (I am guessing at this) to plane on your 5.5. That's where the 6.5 will help - it will get you planing in less wind.
Hope this helps.