Have little trouble believing in both, with some caveats about how the argument is framed.
First there is no conflict, because:
My belief in God doesn't come from inherent trust or distrust about the origins of the Earth either way. It is what is visible in the experiential historical realm, including self and others now, as well as back in times as far as King David, the whole range. In grandiose terms, my belief doesn't rest on the infinitely un-witnessed and un-testified by human eyes distant past, but on things known and that continue to the present. The OP' statement about hardline stances comes within the ballpark of my outlook also.
Put another way, if your belief OR disbelief is having to center on the Earth being 4 billion years old, or not being that old, I have to wonder why either position is held so strongly? Put even stronger, they could confirm the age of the Earth tomorrow as either just over 10,000 years (Atlantis sinking date per legend), or 6,000 or 6 billion, and it would not affect my faith. It is almost in the same category of proving or disproving the Egyptians built the pyramids--- because the reasons are not centered on how convincing the origin story is. (Interestingly enough though, this starting point emphasis was a feature of pagan inquiry in the 1st C---- so some things come around full circle, it seems.)
These days there seems to be a clash between accepted earth and biological sciences, and what is apparently termed YEC (6,000 year Earth) on the board. Yet archeological excavations move dangerously closer back in time to that very mark each day, never mind the 4 billion or so, or even the dinosaur age disputed part. Particularly in Egypt, the earliest traces already show a rather developed character, which of course requires a starting point a bit further back. So each find a ruin that is older still, has bearing here. What are saying is 6,000 years seems a very unsafe mark even in historical terms.
That above said, there are problems as stated at top with how argument is framed.
There has been much said about `assumptions' -- but one of those assumptions is that what was conveyed in Genesis had to mean a certain way, or even was spoken with the same tenor each time. Adults tell kids all kinds of things. The rest falls under grandiose curiosity that can be indulged, but should not govern your life. Where the one G-d, the Great UnManifest, is concerned, there is far too little space left these days for the wider concept of MYSTERY, Mysterion, humility, as it was understood when talking about the Logos.
Today, on the other side there is an almost militant atheist tone, that presents evolution in terms more like affirming abiogenesis. It is how they frame evolution, as compared to how it was done in class when growing up (parents recall the same) where you learned evolution and the dinosaurs, and it was just fine, no implied or pushed contradiction. [It bears noticing that the same school of thought that pushes evolution with this "tone" is the same one that is going around pushing what amounts to a form of religious secular-liberalism, completely with unproven assumptions and even often denials of science where it conflicts with social engineering agendas not unlike what they accuse YECs of.]
If you think of some of the stupid counter-to-sense things being pushed about man and woman and ignoring gender difference realities today - evolutionary traits by the way - you see where some of the friction comes from. What the `anti-evolution' side has to remember is that a great deal of the biologists, geologists, and astronomers that came up with the theories of the age of the Eart were NOT doing it as part of the current PC-liberal religion, and in fact routinely were both believers and scientists. That is, they were not forging results or creating deceptions as a rule. Thankfully the more important foundational sciences pre-dates today's highly polarized age.
I suspect that anti-evolutionists see evolution as something like the extremely dubious and hyped up climate change argument, where routinely colluded emails or massaged findings and models turn up, and where it is all so clearly political redistribution driven.
But geologic and Earth science is not like that -- it pre-dates that period, was the product of a far more sober and well argued out over time process. Once you realize the geological evidence, there is no reason to focus on the dinosaurs existing or not as a basis for accepting the very old Earth. It is also significant that expeditions off-world to the Moon and Mars tend to confirm that planets are very old in relation to our historical perspective.
Evolution in certain basic outlines makes sense, but it is often framed today in a tone that is abiogenesis in disguise. There are some problems have with evolution, but they are more of an intuition kind. In any case, I don't see any reason to put them in such opposition.
It is those who put them in opposition, especially government, activist, and educational curriculum and agendas, I find far, far more suspect than trying to reconcile the evidence of science and imprinted truth of the divine.
