Welcome to the wonderful world of color management.
If you're saving the image with embedded icc-profile, in a format that supports profiles, use a web browser that supports full color management (Firefox but not IE), and have your Photoshop color settings set to "preserve embedded profiles"....then you will not see a color shift.
Whether the color you then see is right, depends on the monitor profile. But it should be consistent.
Otherwise all bets are off.
One that would keep the image's true color?
You might be subject to a misconception; the R-, G- and B-values in an RGB image do not actually describe a color if they are not allocated a position in the Lab- (or like) space (via an ICC-profile).
They might indicate that a pixel is darker, lighter, more red, etc. than another pixel but which (human) color perception it should actually trigger is not clear without the profile.
What are your Edit > Color Settings in Photoshop? (posting screenshots can save one some typing)
I suggest as a first step, check all those boxes in the Color Management Policies section.
Another thing that could help is to open your OS Color Management panel, look in Devices, and post a screenshot of that too. Inevitably you'll be asked what monitor profile you have set up.