Fence panels get blamed for a failing fence and are almost never the cause. Timber rots where wet and air meet, which is the few inches either side of ground level, so that is where a post goes first. Everything above it can be sound while the fence is effectively no longer attached to the ground.
Testing a post
Take hold of each post and push firmly. Sound posts barely move. A post that rocks with the soil moving around it has lost its footing. A post that moves while the ground stays still has rotted through at the base, which is the more common failure and the one that gives way in the first real gale.
Check at ground level with a screwdriver. If it sinks into the timber with light pressure the section is gone, whatever the rest of the post looks like.
Why concrete does not save a post
A timber post set directly into concrete often rots faster, not slower. The concrete holds water against the timber and the post sits in a damp socket for years. What matters is drainage underneath: a layer of gravel at the base of the hole so water can leave, and a concrete collar finished above ground level and sloped away so rain runs off rather than collecting around the post.
Repair without dismantling the fence
A rotten post does not require taking the whole fence apart. A spur, which is a concrete or steel section bolted to the sound upper part of the existing post and set into the ground beside it, carries the load and leaves the panels in place. It is a considerably smaller job than replacing a post and perfectly sound when the rot is confined to the base.
Gates
A dragging gate is usually the post leaning rather than the gate sagging, because a gate post carries a constant sideways load that a fence post does not. Check whether the post is plumb before adjusting hinges. Adjusting the hinges on a leaning post simply moves the problem and it returns within months.
Where a gate is used daily, the gate post deserves a deeper hole and more concrete than the rest of the fence. It is doing a different job.
What helps and what does not
Keeping soil, bark, leaves and climbing plants off the base of a post helps considerably, since all of them hold moisture against the timber. Painting a preservative onto a post already rotted does not help at all, and painting the top end-grain is worth more than painting the sides, because that is where water gets in and travels down.