Fakesharksfinsoup,
OK let's see where we stand.
You said that you could see the Referrals box on the "login" page, but I assume that you mean the "Create Account" section. I am using a split login so the Create Account section is a separate page.
I assume that:
1. You can see "Require Referral" under Admin>Configuration>Customer Details and that you have set it to "True" or "False" as you wish. I prefer False so as not to annoy my customers.
2. You can see "Display Other Referral option" on the Admin>Configuration>My Store menu and you have set it to "True".
Then you should be able to see BOTH the drop down menu for the preset referral sources AND the text box for the "Other" source information.
3. You have "Referrals" and "Referral Sources" as options under the Admin>Extras menu and that the "Referrals" option is populated with a number of suggested sources (e.g. Google, AOL, etc.)
If all of the above is true then you probably have installed the mod correctly, but I would still click through the directories to make sure all of the mod files are installed where they belong.
That's the simple part...
You said you are using ZenCart v1.38a. I just went back and checked the 4 files which I thought might give the most difficulty because of other mods added and there were no real conflicts. If you have the latest version of the How Did You Hear About Us mod they are essentially v1.38a based files. So you should be OK.
Can you give me a link to your site so I can look at it?