As a newbie, I also cleaned my "not so dirty" bike (about 500miles on the bike) once recently via watching Youtube videos and TeamEstrogen.com's "How to clean a bike" advice page.
I used a screwdriver to scrape the grease gunk off of my jockey wheels before using the degreaser. I also took the back wheel off to scrape and scrub the cassette before using the degreaser. Maybe, that's why spraying a ton of degreaser didn't work well??
But, one of the last things to be done is to clean the frame (not the first thing...because as you had noticed, the frame gets dirty again). I used a little bit of Murphy's Wood Oil Soap (which is safe for cars, so I am hoping that that cleaner is also safe for bikes) with a lot of water. Since Dawn takes away grease, I worry about you using Dawn for cleaning your bike frame and stripping away your bike's waxy coating.
Also, I used the Park's Chain Tool cleaner to clean my chain (hook the Park's tool cleaner to the bottom chain and back pedal while your bike is hanging on your bike rack with a large towel underneath). That totally cleaned my chain well.
Lastly, I rinsed and wiped everything down, hoping that I got all the degreaser off of my bike frame. Let things dry. Then I oiled my chain and tried to figure out how to oil the different pivot points. Apparently the chain oil is different that the oil used for pivot points.
I did my best, but I think that I will take in my bike for a once a year full service with cleaning and lubing.
Hopefully, the next time I clean my bike, I'll be faster too.
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2012 Specialized Amira Elite, upgraded carbon handle bars, Jett saddle 143mm switched to 145mm 2012 Selle Italia Max SLR Gel Flow saddle
2011 Specialized Ariel Sport,suspension post,Serfas Rx Women's Microfiber saddle