There is excessive play in my steering and I had planned on replacing my center link, but then I heard this would affect my alignment. Can I simply replace the old worn-out part with the new one, or will I also need to have the car re-aligned?

 

Yes you need to have it re-aligned.
I would take it to Sears.
I used to work for them years ago.
They sould check it for free.
But chances are that it will need an alignment.