A few years back, I participated in a tea swap, and I received a number of Teavana teas. They were all "mixed teas" that had lots of "extras" in them (dried bits of fruit, flowers, other non-tea things). I did not enjoy them, but they struck me as potentially enjoyable with sweetener.
So my first question might be whether you enjoy tea with sweetener (sugar or other)? I also get the impression (again, from what I've heard) that a number of Teavana teas fall into that area. For example, the "chamagne sangria punch black tea" sounds like (to me) one that Teavana would serve with a lot of sugar. It's not good or bad, it all depends on your palette.
If you're okay with no sugar, I would at least expect their 'monkey picked' oolong to come out okay, and so you may be correct that you are oversteeping. My typical routine for oolongs starts with ~1g of tea for every 1 oz of water (boiling). I steep for 30 seconds, discard that steep. Wait ~1 min. Then add boiling water for 1 min -- and that's my first cup. I usually then do additional brews at 1 min, 2 min, 4 min, etc.
But different teas require different brewing parameters. For example the green tea you should probably use cooler water (170-180F). In general if the brew seems too strong or too astringent, you could try reducing the amount of leaf, lowering the temperature, or lowering the steeping time.
We might also be able to give some better pointers if you have the time to take pictures of the tea and your brewing apparatus (for example, I have no idea how much volume the IngenuiTEA holds, and of course the most important part is how much you mean when you say you "eyeball some tea").
What all that text means to say is -- I think we can help you get to better brews with what you have, and then we can help you get to better tea sources.