Have you quoted the right model number? The D3500 although a newer camera would not be an upgrade on your D5300.
I admit to being a chauvinist for the D750. I do use live view quite a bit (the screen tilts so I can reach a slightly higher viewpoint) and focus can be very slow in LV but it is accurate if your bird hasn't flown away before it's finished.
A big consideration for you might be the need to buy new lenses if you go full frame. The D750 and other Nikon full frame cameras
will take pictures with DX lenses but they will be (quite a lot) worse than the ones you get from your D5300. If you haven't the FX lenses already and don't want the expense of buying them, then the D500 is a good suggestion - IIRC it was known as a 'sports' camera, being fast in terms of focusing and frame rate which is probably why the ancient mariner thought of it.
You might also consider the D7500 if you want built in flash which the D500 does not have. The D7500 has essentially the same sensor and jpeg processor as the D500 so should produce similar images all else being equal,
but the D500 should have the better AF as it has more points (the D7500 AF I think is quite similar to the D750's, updated a bit).
Somebody helpfully did
a comparison of the D500 & D7500. Full disclosure, I have used neither of them.
EDIT: Sorry the above crossed with your reply at post 12.