The story of Rama is given in from chapter 47 of narasimha purANam. http://vedpuran.com/brahma.asp?bookid=4&secid=1&pageno=001&Ved=Y&PuranName=Narsimha page 165
Many incidents that are popular today match closely with this version and are quite different from Valmiki’s version.
For eg valmiki’s version does not have a seetaa swayamvaram, but narasimha purAnam does.
In narasimha purANam the other princes are able to lift the siva’s bow but not able to break it, and breaking the bow is the pre-condition to win sitA’s hand. In valmiki rAmAyaNam the story is quite different and I have written about it in great detail in the sItA dEvI page, please find and read using the site map.
In valimiki rAmAyaNam ahalya is a just like wind in her hut. In narasimha purANam she is pASANa or stone, though the incident of rAma’s foot touching the ahalya stone is not there in either version.
In narasimha purAnam the first narrator is mArkanDeya, and sUta lOmaharSaNa who learnt about it from vEda vyAsa is telling this interestingly enough to bharadwAja who heard it first from vALmIki!
It is all very interesting and if you know hindi or sanskrit – please follow the link.