It seems obvious that the problem has something to do with playing too hard, and that you need to lighten your touch, but it seems important to figure out what exactly is hitting what to cause the "thump." Just striking the strings aggressively doesn't cause a "thump" sound. If it's not the sound of your finger hitting the pickup, what it is it?
The only thing I can think of is that you're using rest strokes -- i.e., after plucking a string, your finger continues to move until it comes to rest on the next higher string -- and that the thump is coming from the latter part of that motion: That is, your finger isn't just "coming to rest" on the next higher string, but is smacking it pretty hard. If that's the case, the problem isn't in the string-striking per se, but rather in the follow-through. When you're "getting into it" you might actually be doing a good job of NOT striking the target string too hard -- but at the cost of taking out all that extra energy on the next string. If so, you're not only causing this annoying "thumping" problem, but you're wasting a ton of energy while you're doing it.
As others have suggested, turn the amp up so that a light touch is adequate to produce the volume you want. Practice playing gently this way, and then work on resisting the urge to pluck harder just because you're "feeling it."