I am not religious and have the same issues with some biblical names. I think [name]Noah[/name] is borderline for me - it’s definitely biblical in one sense and just a mainstream popular name in another context. A lot of the biblical names I like share that second quality and don’t seem biblical to me because they are just so popular, like [name]Daniel[/name], [name]Joseph[/name], [name]John[/name], [name]Sarah[/name], while in the meantime, names like [name]Jacob[/name] have a nice way about it, with a slight biblical edge to me that I could overlook. Similarly, you don’t get much more biblical (in my perspective), than [name]Adam[/name] or [name]Eve[/name] - I consider them both usable. I think [name]Noah[/name] has that property that I would consider if I liked it as much as you do. Pass on names like [name]Elijah[/name] or [name]Isaiah[/name] or [name]Moses[/name], for example, or references like [name]Evangeline[/name] or [name]Seraphina[/name] (I am arbitrarily ok with [name]Angela[/name] & related names).
I would take the bible source as similar to a nationality - I am not French in heritage, but there are mainstream French names I am comfortable with. Alternately, I am part [name]German[/name] and Italian by heritage, and the same is true - some comfortable pool of imported names that I would feel ok using, and a very big lot of names I would consider unusable in the US unless you just immigrated and really have an authentic reason for considering it. Some biblical names clearly seem to have more meaning to people and are outside the mainstream, while some are so common there’s no real barrier to use them, no matter how you feel about religions.
As for popular names, I don’t know that you can help what you really love. You might rather theoretically prefer a name to be less popular, and some people are strict enough with their requirements that they would discard a name they love on that basis, but I think if you have a good name that you want to use, you shouldn’t let that stand in your way, and your child would probably rather have a name you love than something that appeals more to the vanity of being singular - to you. That might not matter to them at all. I would rather my parents love my name than pick something that was more of a second or third choice or a bigger compromise between parents who couldn’t find anything they both love as much as the #1 choice. Sometimes a name is just that good, and popular because it is so appealing. You don’t need to fight it!