One of my rules of thumb is "When multiple pieces are attacking a square adjacent to the King, watch for mate threats." After 28. ... Qxe4, there are three heavy pieces attacking g2. You should have defended with 29. Rd2.
bungled attack
This is a game I played here yesterday. I think I had a win but missed it. I have tried to show it in the variation beginning at move 27 although I have included other variations earlier.
I'm sure there are other possibilities but I haven't found any better.
You suggested 27. Qa4 Qc6. I agree that 27. Qa4 is a strong move but I don't think black would play 27... Qc6 since that looses right away as you showed in your annotations. Playing 27... Nd4 sacrificing the knight would be better. Black is loosing a piece anyway so he might as well get the most out of it although white still has an attack going on (28. cxd exd 29 Ra3).
One of my rules of thumb is "When multiple pieces are attacking a square adjacent to the King, watch for mate threats." After 28. ... Qxe4, there are three heavy pieces attacking g2. You should have defended with 29. Rd2.
29. Ne3 is better as far as I can see. Protects g2, attacks the rook on g4, and black's knight on b5 is still under attack. It also rescues the knight since it was under attack from the queen on e4. So although it removes the knight from attacking black's king it does look good to me. Did I miss something?

On the contrary, BlueKnightShade, you saw something I hadn't. 29. Ne3 is good; I hadn't thought of that because I so badly wanted that d6 pawn. With the knight adding to the defense, Black will not be able to sustain a winning attack at g2. If Black moves the rook to safety, the knight is lost. A classic case of too many attacks at one time.
I also like 29. Rd2 because Black will either want to move his knight to safety when the d6 pawn falls with attack on the queen, or else he will want to continue the attack at g2 which wins the equivalent of two pawns after 29. R3d2 Rxd2+ 30. Bxd2 Rxd2+ 31. Rxd2 Qxc4. At that point White has queen and rook against queen, knight and bishop and is down two pawns. It seems in that case that Black stands better.

notsyor38, that would also work well: after 29. ... Qxe4 White can drop his rook back to play defense (30. R6d2) and if Black presses the attack with 30. ... Rxg2+ 31. Bxg2 Rxg2+ 32. Rxg2 then we find White with a colossal material advantage (roughly two minor pieces). And if Black chooses something other than 28. ... Nxd6 it isn't going to be much to choose from that I can see.
This is a game I played here yesterday. I think I had a win but missed it. I have tried to show it in the variation beginning at move 27 although I have included other variations earlier.
I'm sure there are other possibilities but I haven't found any better.