Problem with colors on exported vid

You didn't have options for color space for video before. There used to be only Rec.709 for a working color space, and all video was Rec.709. That time is long gone.

Now there's Rec.709, and several flavors of HDR ... HLG, HDR10, HDR10+, PQ,  and DV ... DolbyVision.

Premiere can work either set to Rec.709, HLG, PQ, and if you know how, HDR10, or to pass through without any CM operations for high-end work like vfx plates and such. And you have to choose, as Premiere can't "know" what you want to do.

So you probably would be better suited to a Rec.709, and pretty automated color managed workflow. Easy peasy.

Lumetri panel Settings tab ... the tab NAMED Settings.

- Set Display color management on, and if on a Mac, probably Extended Dynamic range also.

- Set Auto Detect Log and Auto Tonemapping both on.

- Set the Sequence working space to Rec.709.

That will probably handle most things for you. With one ... other  ... issue. Viewer gamma setting. 
So first, we have a question ....

Are you on a Mac, without Reference modes for the monitor?

If yes, read the next few paragraphs. If no, skip to Viewing Gamma Settings below.

Here's a choice that's a total pick your poison. Why?

Apple uses a wrong display transform for Rec.709 media but only on Macs without Reference modes. So if you are on one of those, then outside of Premiere, but only in QuickTime Player, Chrome, and Safari ... you will see a lighter image in the shadows, with less visual saturation and slightly shifted hues.

All other screens see the more normal representation of Rec.709.

If on a Mac without Reference modes, check your video in both Qt Player and VLC or Potplayer. Everyone not on a Mac like yours will see the VLC/Potplayer image. Those on Macs without Reference modes will see the lighter one.

There isn't a fix, as you can't show the same image file with two different display transforms and see the same image. Thank you Apple for messing things up so bad. Ah well.

So do you only care about what it looks like on Macs like yours without Reference modes?

Then set Premiere's Viewing gamma to 1.96/QuickTime, and correct your video. Outside Premiere it will look similar but only on Macs without Reference modes, while viewing in Qt Player, Chrome, and Safari.

ALL other screens will show a darker, more saturated image. And one other comment ... 

No professeional distribution media is ever graded for display transform of gamma 1.96

Viewing Gamma Settings Explained

- Gamma 1.96/Qt Player sets the Program monitor in Premiere to use a similar display transform for Rec.709 as to what Apple's ColorSync CM utility does. As stated above, the image will appear similar within and without Premiere, but only on Macs without Reference modes. And even then, only in Qt Player, Chrome, and Safari.

- Gamma 2.2/Web ... This should be the standard setting for most users!!!!!

NOT because you are grading "for the web" ... but because, by the professionally set and followed standards for pro video color correction ... IF you are working in a normally lit room, the display transform should be gamma 2.2. 

- Gamma 2.4/Broadcast Should only be used when grading in a very dark, not quite blackened, room, with a proper bias light on a gray surface behind the monitor.

If you're not working in a pretty darn dark, gray room, do not use gamma 2.4.