Sounds like you have a disproportionately long torso. Truth be told, you're going to have some challenges getting a bike that fits and handles well for you.
Part of the issue is that hardtail mountain bikes <$1000 are nearly all essentially based off of cross country race bikes, where having a relatively low handlebar position combined with a relatively long stem is intended to keep weight over the front wheel during steep climbs. This is unfortunate for riders who do not ride at that same intensity and may not have the flexibility for a lower riding position.
Standover varies a fair amount between bikes, and to a certain point is a matter of personal experience, feeling, and experience as to how important it is. I have a moderately disproportionately long torso (I'm also 5'11"), but I'm also an experienced, relatively fit, rider in my 30s, and so I'm not particularly concerned with standover so long as I don't think I'll actually injure my genitals during an unexpected dismount. I touch on my road bike, on my full suspension bike it is sometimes awkward to mount with the saddle up , etc. For you standover is understandably more important.
Maybe consider a bike with a dropped top tube--this may be marketed as a women's bike, but get the bike that works for you!
Also, you said you have the saddle all the way down. Is this to make mounting/dismounting easier or is that a height that gives you a full leg extension with a slight bend in your knee (the correct pedaling height)? If its at the right height for pedaling with the saddle slammed, and you're 5'11 on a medium frame, you have crazy off the chart body proportions and will have to either compromise in some significant ways or have a custom frame built.
As for handlebar height--hand pressure is in part a function of core strength and power output. If you put sufficient power into the pedals, you will actually be pulling up on the bars. The lower your core strength and power output, the higher your bars need to come. The best ways to raise your bars are to use a high rise stem and/or a high rise handlebar. Note that if you need to raise the bars significantly, you will need new cables and housing. If the bike has hydraulic disc brakes and they need to be lengthened, the whole hose will need to be replaced and the brakes rebled, which can lead to a relatively expensive service bill. The disadvantage of raising your handlebars or shortening your stem in this case is that it will speed up the steering of the bike, possibly to the point of being twitchy. I'm a professional bike mechanic and I test ride a lot of customer's mountain bikes that have very high handlebars, usually bikes by older riders who mostly use them on the street. You can get used to a lot of steering characteristics, but these bikes ride really twitchy, and I would not prefer their handling on a single track trail.
I hope you get things sorted out to get you comfortable on a mountain bike!