I don't understand certain things about your post.
You seem worried about openings that are "drawish," but nothing you name is drawish below a certain masterly level. Yet if you were worried about stats at and above the masterly level, surely it's worse that 2.Qh5 scores over 57% for BLACK at that level?
If you want aggressive, sound, and no gambits, the Scotch is a good way to go. As is the Center Game. (Which is equal but scores better for white all the same, but that's more than you can say for the Parham.) Ditto for the Bc4 Vienna. Or the Bishops game featuring a repertoire with lots of early f4's, as recommended by John Emms in his aggressive white club repertoire book.
If e4 isn't your bag, lots of good choices with d4, c4, or even f4 that would meet your criteria of being sound, aggressive, non-gambity, and better than the Parham.
Recently, I have been in arguements in nuerous threads with many of you about the Parham Attack. For those of you who don't know about it, here it is:
1. e4 e5 2. Qh5
To most this seems like an amateur move, but I believe it is much more, and has even been employed by high level players (Hikaru Nakamura, and Bernard Parham who always played it). Many of you have tried to convince me this sucks, but as you know, I full-heartedly believe it is good.
So here is my question to you, is there a better opening? Here's what I want out of my opening:
1. Fairly Aggressive
2. Not forfeiting material (so no gambits)
Are there any openings you know of that meet those specific requirements? The Parham meets both of those for me, but you all still insist it is bad, so give me something better that meets those requirements.
Two openings I like are playing 4. Ng5 against the two knight's defence, and playing 5. Nb3, against 4... Bc5 in the scotch. However, the alternate popular lines, such as the giuoco piano after the italian game, and the mieses variation in the scotch game (4...Nf6), are drawish, and I have trouble meeting those aggressively but sound. If there are anymore direct ways to reach these, where there aren't many alternative lines, those would be very much appreciated.
So to conclude, can any of you provide an opening that is aggressive, without giving material? Two I like are 5. Nb3 in the scotch, and 4. Ng5 in the two knight's defence, though I don't have a direct way to reach those.