$4 NLHE 6-max: Played KK too passively?

Before I get to the hand, a brief set up.

I've been on a losing streak for 3 weeks and have had some disheartening beats so I think it made me real timid here.

Minutes before the KK hand, I had hit a set of 6s on a 2h5s6s flop and gotten it all in 3 way showdown vs. 4d5d and 9s9h, lost to a 3 on the river. I was still feeling pissed off from this when I got dealt KK.

I had $3.41, villain (SB) had me covered.

I was second to act, I raised to 14c. Villain 3 bets to 46c. Normally I just shove in that spot, but I saw that his 3bet% is 1 over 500+ hands, so I just called thinking it's at least AK and likely AA (I had run into AA when I had KK or QQ 3x in the last few days before this).

Pot size: 96c

Flop is 679 rainbow. He bets half the pot. I just call.

Pot size: $1.92

Turn is an 8 and brings a flush draw. He checks, and I checkd.

River is an Ah. He bets $1. I think and fold.

I feel like I played real scared. How should I have played this better?

4bet then shove flop and worry about AA when he flips it over. I say this mostly thinking that 500 hands is't enough to make the decision and because QQ and maybe AK are probably in the average persons range even those on the tighter side.

4bet pre($1ish) with intention of getting it in preflop.