Go ahead and get the oven thermometer. In my professional opinion, everyone should have one anyway as part of their standard kitchen equipment.
Odds are, this is one of the reasons why commercial cake mixes are so popular, as the average cook doesn't have to worry about what kind of flour are they using, humidty/temperature/static on the day they bake, etc. Just add the milk/oil/egg called for, and the cake will turn out.
Personally, I don't use cake flour, myself. I rarely ever bake, and we have such a problem with weevils that that cake flour would be badly infested with them by the time I actually used it all. There's a few cakes that I make completely from scratch, and yes, they are a little denser than what I could get out of a box. I even have one recipe for a cake from scratch, that the original author made a comment that this cake is "the usual country-style cake, rich and very dense." This leads me to believe that a lot of the cakes made from-scratch in-home where nowhere near the soft texture that today's commercial boxed mixes can produce.
In the end, you'll have to make a decision. Pay the extra money for a different kind of flour, and see if that gives you the results you are looking for; or just accept what you can make.
In all fairness, while I don't bake that much, when I do bake, my kids, spouse, and MIL don't care about the texture, they like the flavor of my cakes.