Should I Let the PT Fence Dry Before Installing

A concrete post/form in the ground with some sort of above ground post or embedded post in the concrete is the way to go. Depending on what you want from your fence, ie: privacy, security, looks, etc. will effect what kind of materials to use for the actual visible fence. There are many choices of composite materials and vinyl products out there that are low or no maintenance and will outlast any wooden materials. The first serious structure to fail is usually where a wood post meets the ground. You need to pay special attention to this area, keep wet dirt, leaves etc from building up around the wood posts. The concrete should be sloped away from the post so water drains away and no water stands around your posts. Use good hardware, galvanized at minimum, stainless steel screws are best, but a little pricey.

I'm not sure if either contractor has really given you good advise. Tell us a bit more of what you what your fence to do and look like and we can help you more.

Unless you have a kiln to dry wood in, drying wood in a standard environment takes a really long time (if you buy firewood, usually you want ~2yr old wood!). I don't know that leaving it out for a couple weeks would dry it if it were so saturated that there is visible water coming out of it. It is recommended to let wood used for hardwoods sit a couple weeks to get accustomed to the environment as far as humidity goes, but it is not being dried any further.

I would recommend that you find a better source of wood that is dried properly. You also want to carefully select your wood, it might be that you just got a bad piece some how.

For outdoor projects, instead of PT lumber you might opt to try a wood like cedar which naturally resists the elements pretty well.