I was thinking of running a classic contest model where people on our email list would be entered into a drawing for a prize. If they share the contest and their friends enter the contest, the sharer would get an additional entry for each referral. I just need to know of a good platform to run this - the referral program. Or if anyone did a manual setup before, how did they do it? Everything I've seen is at least $50 - $100 a month and we would not be doing this type of contest that frequently.
As you mentioned yourself, there are almost no such referral programs that gives you free services, especially if it is for some kind of revenue generation.
Mostly they give a free monthly trial so you can get used to it and then start paying.
However, here is a list of some referral programs that you might find useful.
In this post you can find a lot of information about different referral programs, their pricing, trial period, Starting fee/month and reviews by users.
As you also mentioned about a manual setup here is one similar program called Prelauncher. Its code is available on Github as well which you can find here.
Referral Candy works very nice and you get access to all of the features in the trial so you can test it for 30 days and see if it works well for you. Also, before the trial is up they will email you to let you know it's almost up instead of sitting quiet and charging you a subscription under the table (like many app companies will do).
I'd highly recommend at least giving them a try for 30 days.