Fixing wood fence posts, 2 questions

I have a wood picket fence, 44" high, with about 35 fence posts, round,
4" in diameter iirc.

I have two problems.

1) One is loose. It feels like it has broken off maybe 8" above the
bottom end. How to fix it????

I thought I would disconnect it from the fence, pull it out (which I
think will be easy), take a replacement (easy to find), cut it to the
same length as the piece that comes out**, put the post in plus some
instacrete or sackcrete***, pour in some water, and let it harden.

Is that okay? If the part of the first post every rots away, and the
post sinks, I can replace it again.

**Well actually I would put the new post in and use it like a pile
driver to crush the old post stub as much as I can. Then extend a
string from the posts to the left and right of this one, and mark this
one so it's the same height as the other two.
*** (the speedy stuff or the regular stuff both work, I hear)


2) Some of the posts are hollow in the top center, going down 2, 3 even
maybe 4 inches. . Water sits on them and the wood rots or crumbles.
Sometimes green plant life grows in the holes

a) I thought I should drill a hole in the side so the water drains out.
or
b) I should fill the holes with something, something that looks like the
original post if possible. . I had tried outdoor wood glue mixed with a
lot of sawdust. but that broke up.

Suggestions?

Thanks.