The thing about kava grog (here I use the word "grog" to refer to the beverage made with kava root) is it's a social beverage and you typically drink several cups of it in the form of a session. These cups aren't full 8oz cups, but smaller amounts, around 5-6oz. You can have bigger cups if you want; however much you can think you can chug. You want to drink a cup, let it settle for about 15-20 minutes, then drink another cup. You make enough kava for several cups over the course of an evening, typically. Traditionally, grog is a social beverage, so you make a large batch to share with several people, and everyone is having multiple cups. In-between each cup, everyone talks, chats, or participates in some type of activity like singing, or perhaps you watch a movie or the game. You can drink kava all by your lonesome but there are somethings about kava that you experience best when you're living the moment with others.
When you drink grog over several cups in the course of an evening or the day (a session length is variable), the effects of each cup layer and refresh each other. The anxiolytic and euphoric parts of kava have quick onsets and wear thin over an hour or so, which is part of why you drink another cup to feel more euphoria. The sedative effects have a longer delay and duration. The heady components of kava have a short duration, so if you're doing one cup, you're not going to feel these for very long. At large quantities, a lot of heady kavalactones can do the opposite of what you'd want. This is where heavy kavalactones come into play as they can tone down higher concentrations of the heady stuff, which could explain your "bit of anxiety in the pit of your stomach."
With having several small cups over time rather than one big gulp in mind, your dosing may be a bit high for starters. I don't know if you've experienced reverse tolerance or not, but at those dosages, it sounds like you're feeling something at least. Try using 4-5 tablespoons of medium grind powder with a pint of warm water, knead for 10 minutes, then get slightly less than a pint of warm water and knead with the same root powder again for another 10 minutes. Then mix the two batches together and drink a cup every 15 minutes or however frequently you wish.
I hope this all helps!