You say that since you're religious you don't like the idea of mixing what is fiction with what is fact.
If one were mean one could say that you're doing this already since your god is fictitious.
But I'm not mean so I won't say that! :-p
OK, on to your question.
There are plenty of fantasy novels that have religious themes, in some of them religion is central.
There are religious authors writing fantasy weaving their particular theology into the novels; some do it rather obviously, such as with the Narnia books by C. S. Lewis, Frank E. Peretti's novels (e.g. the Cooper Kids series, and his This Present Darkness / Piercing the Darkness) or Orson Scott Card's Mormon related fantasies, some keep it a bit less obvious, such as Ted Dekker (who says that he tries to appeal to non-religious readers by not hitting them over the head with too explicit religious content) or Stephen Lawhead.
There are also religious authors that include fictitious religions in their books, such as the Tracy Hickman (of Dragonlance fame) and Brandon Sanderson, both Mormons*, or Gene Wolfe.
For your specific purpose, I'd recommend you to read some of Tracy Hickman's books that he co-authored with Margaret Weis. These two were fantasy powerhouses back in the 80s and 90s; they seem to have been pushed aside in recent years as their work is more traditional and not the currently favored grimdark version of fantasy.
But they do exactly what you are asking for: integrating religion into fantasy.
Their best know works are probably their Dragonlance books (their series started the novel franchise) and I can recommend these are there are gods at play.
But a lesser known series by that duo which is absolutely perfect for you is their "Rose of the Prophet" trilogy.
It's set in a world where you have various deities that each are positioned on a icosahedral system of characteristics. (That sounds very weird but it's not that important now and is less complicated than it might appear.) Many of these gods are echo real-world religions.
* This starts to look like I'm trying to discreetly push Mormon authors at you. That's not the case. I am in fact an atheist but I love fantasy that's why I'm familiar with those names which are pretty big in the fantasy community. I once was Christian for a while which is why I'm familiar with the likes of Peretti.