How to make a thick coffee with an AeroPress? I end up with a watery drink and an over-brewed aftertaste

The easiest way to make the texture of your coffee thicker would be to increase your brew ratio - use more coffee and less water while brewing.

Also don't be scared to use higher temperature water, even if the inventor recommends ~80°C water, this is also going to increase extraction, leading to a higher percentage of dissolved coffee in your cup.

You might also want to look at this "Coffee Compass" from Barista Hustle, which shows you what adjustments to make based on the taste of the coffee:

Honestly, I recommend trying the AeroPress manufacturer's instructions, both hot and cold brew recipes. Then tinker from there.

The cold brew recipe produces less bitter coffee with more crema. Give it a try.

Also see their FAQ for tips and techniques. E.g. they recommend a paper filter for taste and health reasons.

There's really no need for inverted brewing. If you want to brew a long time, you can do that right side up. Compare the taste and post notes about your results.

Thanks to immersion brewing, AeroPress is relatively insensitive to grind size, but a finer grind will produce a bit more concentrated coffee (so I use a level scoop instead of a rounded scoop) and might take longer to press. Pressing too quickly can produce high back pressure.