Narrowed choice down to the Fuji Newest 2.0 and Schwinn Le Tour GS.?

These are very similar bikes. I think that you have already answered the question yourself when you say that you like the look and feel of the Shwinn better. The adjustable stem would not be an issue as far as stability , but if a shop fits you properly, they would normally instal the stem that is right for you, it is only a 5 minute job to change a stem these days, good shops do this as a matter of course. It may be that the schwinn you looked at already has the right size stem for you and that is one reason it feels better. Either way, looks like a good deal.