Anything you begin to paint won't look as you hoped against the existing color. Sometimes when making a drastic color change you have to paint the room in faith that it will look like the sample you picked. Grey will look bluer against golds and oranges. Pick a warmer greige like Benjamin Moore Revere Pewter.

Revere Pewter

if revere pewter seems stronger than you'd like, have a look at balboa mist.

revere pewter above, balboa mist below

You can do "Amazing Gray" by Sherwin Williams; it blends brown and gray together for a warm feel. We are working on a project now with those colors and I think it turned out great!

You might also look at Edgecomb Gray from BM...a little lighter than Revere Pewter.

agree with edgecomb gray or revere pewter at 50 or 75%.

It is difficult to put warm wood tones and grays together - especially in rooms with little light. The best ones usually are the gray blues or the gray greens with wood - and from what you have posted, I think you do not wish to go that route. Give it a shot with some sample bottles with those colors advised above, but if it is not working think about warmer creams.

From last weeks "Fixer Upper" - lots of wood, not sure of what the wall color is.