Deck Sagging - How to fix?
There may be code differences between my area and yours, but beams bolted to the post as in your picture #1 and #3 are not allowed here.
Recommended connection is to notch the post and rest the beam in the notch, or at a minimum use metal beam supports bolted/lagged to the post below and through the beam (typically the beam support bracket would be for a non-load bearing connection like for an awning and not the deck floor).

If your post is rotted, you really should replace it along with the other rotted members. The joists themselves in your photo are showing early signs of water saturation which leads to rot and those will need replaced soon if not already. I recommend checking all your framing members for rot and either replacing, or bracing all of them with the understanding that you'll need to replace them or your entire deck soon.

If all you're worried about is replacing a few beams, you can use hydraulic jacks to support the old beam as you do so. Check some online resources for proper post notch and beam installation, they show how to do this in the middle of a span as well as at corners.