Can I split on two characters, running awk only once?

Actually the field separator you define usung -F is a regular expression. Thus the following command should deliver:

ip a | awk -F'[ /]+' '/inet / {print $3}'

ip a | awk '/inet / {FS="/"; $0=$2; print $1; FS=" "}'

The first matching record is split into fields according to the default FS (space). Then a new FS is set. When we substitute $0=$2, splitting is done again according to the new FS. Now $1 contains what we want, we print it. Finally we set FS for the next matching record.

For this very specific case, have you considered a better use of ip? For example:

ip -j -p -f inet a | awk -F \" '/local/ {print $4}'

This will print ip address as a JSON object an search for the local key, which happens to store the IP address. Is is even sharper you can use jq:

ip -j -p -f inet a | jq '.[].addr_info[].local'

I recommend this last command as it will not suffer from changes in ip addr output. However, if you really like your initial design, I would go with:

ip a | awk '/inet / {print substr($2, 1, index($2,"/")-1)}'

or

ip a | awk '/inet / {split($2, addr, "/"); print addr[1]}'

In the first command, we use index($2, "/") to find where / is in $2 and then use substr to generate the substring. In the seconds one, we split $2 on / and store it on addr array.