I'm sorry - I'm really struggling with the wording of this question, so I'm not sure what you're asking.

Something is either true, or not true.

You can make up beliefs but you can't make up facts, so I don't know how to accept science (fact) and embrace spiritual beliefs (fiction).

"The good thing about science is that it's true whether you believe it or not" - Neil Degrasse Tyson

I think we are hardwired for bias. It's easier on the brain.

Are there religions that embrace the findings of science? Of course there are. And about half of all scientists believe in God. The two are not necessarily mutually exclusive.

I have no intention of "embracing" any form of religious or spiritual nonsense.

What is this realm of subjectivity? Subjectivity by it's very nature (not objective) is based on biased thinking. Objectivity and subjectivity are polar opposites.

I don't believe it is possible to remain completely un-biased about that which is subjective.
As for the second question, most definitely no.
Other people's spiritual opinions or faiths mean nothing to me because I know
they are wholly invalid.

That's an epistemological question And uncannily enough, people in the world of social science have given up trying to reconcile objectivism and subjectivism... if you don't count pragmatists, who are basically saying 'stuff it, I don't care about epistemology as long as I get my answer'.

We all have bias because we all have experience. I think the only areas we might not have bias are those where we literally don't have any previous experience in any capacity, so we haven't formulated an opinion yet. When it comes to science and religion, I don't think anyone can truly be unbiased. Speaking for myself, I wouldn't want to accept faith-based claims, because evidence is too important to me. I definitely have bias against anything that's unsubstantiated, and I think everything in my experience justifies that bias — because without evidence I have no reason to think any claim is valid, no metric by which to guage truth or likelihood. If someone claims Sasquatch trashed my yard last night, I want evidence. If someone accuses me of a crime, I demand evidence. If someone claims a miracle saved them from dying in a car wreck, I want evidence. I'm fine with philosophical inquiry and thought experiments, but claims require evidence, and I won't accept or embrace a claim on the basis of wishful thinking.

The former yes, to an extent. As long as there's something you don't care about, the element of subjectivity is almost negligible.
The latter no, it's an inherent conflict.

But then again having an opinion on this post means I am inherently subjective about it.

Not trying to be a dick, but I have zero interest in embracing such things. I draw the line at "accepting".