Or, put another way, don't throw your approach away too early...
But don't be afraid to throw it away either, and at certain aerodromes - throwing it away early will save your life! If you're not dead on the speeds, straight along the centreline and your wanted glide path, and you are not confident you'll make it - there is absolutely no shame in going around.
I think the best thing you can do is use your watch method, and keep practising. ( [difference between runway and wind direction] divided by 60 ) times wind speed. 250/10 on runway 21 means 40 degrees difference. 40/60 is the same as 4/6 which is equal to 2/3. 2/3 of 10 is roughly 7kts and that is roughly your crosswind. You don't need to use exact figures... If it was 235/10 on runway 21, thats 25degrees difference, 25/60 is roughly 30/60 = half, so the crosswind will be slightly less than 5kts. Keep coming up with random numbers and random runway headings over breakfast, lunch, dinner, whilst watching TV...etc... and keep doing them whilst eating / watching TV until you feel confident with it. (the multi-tasking helps to make the figures easier when overloaded during flight).
With experience, although if you intend to go commercial, you will need to do them before every landing, you may not need to know the exact figure, just to be aware which side the crosswind is. 17kts is the "demonstrated crosswind limit" of the PA28, with experience and skill however, one's limit may differ from the book figure (providing insurance and club rules don't prevail of course).
Keep at it - they'll become second nature! That'll be really useful for your GFT (or whatever it is now called).
Good luck with your learning!