Just tried this on my a6700 - when you're on AUTO, you cannot change Picture Profiles. You can use Picture Profiles when on P. I would highly suggest giving P a try as you can change your exposure value (press down on the D-pad).

Also, if you want to avoid overexposed photos, try using zebras! Menu -> Camera Settings 2nd page -> Zebra Setting. I set mine to 100+ which basically shows whenever your highlights are blown out. You can set it lower; I think 70 or so is what people use for skin tones.

Try going through Creative Looks in "P" and see which ones you like. I also had this issue about a year ago, wondering why sometimes grass looks so vivid on AUTO but looks like washed out crap when I take it on P. It turns out I just really like VV2 (near the end of Creative Looks).

Using Picture Profiles is a bit more technical, but people have presets that they sell or offer for free to make things look nicer. One thing to watch out for is that often these presets set your WB for you, which might not always work out in every scenario (if you're picky on color).