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I think you're doing the right thing. I was also very careful with my first straight when I got it, not more than 15 strokes.. today, I do closer to 50-60-70 (don't really count anymore). Take it easy and with time, you'll feel more comfortable and confident in your stropping strokes. You'll find yourself increasing the number of passes and you'll also feel the difference on your face, when you actually start improving the edges with stropping.
Linen is a good idea for drying a blade's edge, and also after honing, to clean the metal residue, so it doesn't end up on your main leather piece.