Dear Carol, how are you? Firstly, I want to thank you for your great question.
Any plumbing problem can be resolved, however, the question that needs to asked is this: how easy is it going to be?
Unfortunately, shower drain is one of those drains that it will take a bit of work to replace it and, unless you have done a similar task before or know someone that can do it, you will have to probably contract a profesional to take care of it.
You seemed pretty sure where the leak is coming from, so, if that's the case, it is possible that the drain leaking from the joint fittings or that, the drain is simply broken.
Now to address the problem, you will have to ask yourself two questions: How accessable is the shower drain piping? How damage is the seiling below the drain?
Most shower drains won't have an access from the top, which means that, no matter whether the seiling is damage or not, your best option will likely be to cut the seiling to expose the piping and to fix the problem correctly.
I sorry to give this bad knews, but this simply what might have to be done. So, as I said before, this project must be done by some who really knows what he or she is doing.
And, to give you an idea, if a professional does it for you, it will cost you anywhere between $750.00 to $1500.00.
Please, visit my website page for more information about how to replace shower drain at: http://www.diy-plumbing-and-drain.com/shower-drain.html
I hope that I was able to help clearify your questions and, please, if you have anymore questions, just send me an email through my website.
Remember, the information above is for informational purpose only, any project you do, you do it at your own risk, always consider consulting a professional before starting any plumbing project.
