I do agree that AdSense ads will create a lot of mess on your site if they are not properly setup or are on by default settings. However if OP can manually place the ads, that will not be bad looking and the site will still look clean and tidy.
I'm really not sure what kind of mess Adsense is making, considering that they are one of the few (from low & middle-tier ad networks) that have the possibility of automating the size of ads. Just use the choice of responsive ads, where the size and format will automatically adjust to fit available ad spaces and screen size.
I have seen a few websites where they place a lot of ads and the site ads are more than the content on the web pages. People think that if they place a lot of ads, they will get more clicks and hence more revenue but in fact it will just distract the people and eventually people will move away from the site.
OP may have similar issues and other thing is that if they use any WordPress extension that will automatically place the ads on the site, such options aren't good either for the readability of the site.
