I'm answering my own question here partly to help others with this problem, and partly as an archive for myself so I know how to fix this issue if I come across it again. I spent quite a while trying to figure it out, and wrote detailed instructions, so see below:

Installing cartopy is a huge pain, and I've found using conda to be a very bad idea (it has bricked itself and python along with it multiple times for me)

THIS INSTALLATION IS FOR LINUX.

Step 0. Update apt:

apt update

Step 1. Install GEOS:

Run the following command to install GEOS:

apt-get install libgeos-dev

In case that doesn't do it, install all files with this:

apt-get install libgeos-dev libgeos++-dev libgeos-3.8.0 libgeos-c1v5 libgeos-doc

Step 2. Install proj dependencies:

• Install cmake:

apt install cmake

• Install sqlite3:

apt install sqlite3

• Install curl devlopment package:

apt install curl && apt-get install libcurl4-openssl-dev

Step 3. Install Proj

Trying apt-get just in case it works:

Unfortunately, cartopy requires proj v8.0.0 as a minimum, but if you install proj using apt you can only install proj v6.3.1

Just for reference in case anything changes, this is the command to install proj from apt:

apt-get install proj-bin

I'm fairly sure this is all you need, but in case it's not, this command will install the remaining proj files:

apt-get install proj-bin libproj-dev proj-data

To remove the above installation, run:

apt-get remove proj-bin

or:

apt-get remove proj-bin libproj-dev proj-data

Building Proj from source

So if the above commands don't work (it's not working as of 2022/4/8), then follow the below instructions to install proj from source:

• Go to your install folder and download proj-9.0.0 (or any version with proj-x.x.x.tar.gz):

wget https://download.osgeo.org/proj/proj-9.0.0.tar.gz 

• Extract the tar.gz file:

tar -xf proj-9.0.0.tar.gz

• cd into the folder:

cd proj-9.0.0

• Make a build folder and cd into it:

mkdir build && cd build

• Run (this may take a while):

cmake ..
cmake --build .
cmake --build . --target install

• Run to make sure everything installed correctly:

ctest

The test command failed on one test for me (19 - nkg), but otherwise was fine.

You should find the required files in the ./bin directory

Finally:

• Move binaries to the /bin directory:

cp ./bin/* /bin

• As per Justino, you may also need to move the libraries:

cp ./lib/* /lib

Now after all this, you can finally install cartopy with pip:

pip install cartopy

After doing this, my cartopy still wasn't working. I went home to work on this next week, came back, and all of a sudden it was working so maybe try restarting

The libraries should be copied manually sudo cp ./lib/* /lib This works for me