Shell scripting: Select folder based on part of file name
How about using awk with the field separator option -F  and separate the field by the hyphen. Then get the third field.

UPDATE 

I have updated the code to use the result returned from the awk to place the destination folder. This saves on a lot of code. And also used the separator " - " as Ian C pointed out in the comments. 

#!/bin/bash

# Go to the Completed Projects folder.
cd /Volumes/communications/Projects/Completed\ Projects/

# Find a folder with a specified string (e.g. "Academics") in its name.
# Move (not copy) the folder to its corresponding sub-folder of the Archived Projects folder. (e.g. /Academics)

for folder in *; do
    if [[ -d "$folder" ]]; then
        thirdfield=`echo "$folder" | /usr/bin/awk -F ' - ' '{print $3}'`;
        echo "Archiving $folder to Archived Projects → $thirdfield...";
        mv "$folder" /Volumes/communications/Projects/Archived\ Projects/"$thirdfield"/"$folder"    
    fi     
done


I have also added /"$folder" at the end of the move so the folder itself is moved. you can change this if thats not what you want by removing the "$folder" from the end of the mv command.



You can also cross check against an array of the 7 names so only those folders that correspond will be moved. ( you can insert an else statement where needed)

#!/bin/bash

# Go to the Completed Projects folder.
cd /Volumes/communications/Projects/Completed\ Projects/

# Find a folder with a specified string (e.g. "Academics") in its name.
# Move (not copy) the folder to its corresponding sub-folder of the Archived Projects folder. (e.g. /Academics)

# Array of names to check against
ArrayName=(Academics Admissions  Alumni Communications Development President Student)

for folder in *; do
    if [[ -d "$folder" ]]; then
        thirdfield=`echo "$folder" | /usr/bin/awk -F ' - ' '{print $3}'`;

        for var in "${ArrayName[@]}"; do
            # Only move the folder if its key name exists in the arrary
            if [ "${var}" = "$thirdfield" ]; then
                echo "Archiving $folder to Archived Projects → $thirdfield...";
                mv "$folder" /Volumes/communications/Projects/Archived\ Projects/"$thirdfield"/"$folder"   
            fi
        done
    fi
done