I shared your concerns, but have had no problems...
I also had used Dropbox extensively, and wondered why I also needed Dropbox. Let me offer 3 reasons:
1) Let's say you are doing a home renovation and are shopping at various places for lighting. You see an example you like, so you snap a picture with your phone. You also snap a picture of the UPC code that has the price and specs. A helpful salesman appears and you want to record what he is saying. You have a voice recorder on the your phone. You also want to type a few notes, and your phone has a note-taking app. The problem becomes the photos are stored will other phones. The audio note is now stored with other audio notes. Your typed notes are with other notes you have typed. When you get home, you have to remember to find each of these items and transfer them to your phone, create a folder to put them in, and put them into Dropbox. With Evernote on your phone, you create a new note. You take the pictures, record the audio, and type the notes all in the same note. When you get home, the same note is automatically on the Evernote desktop client.
2) With Dropbox, you view anything from your mobile devices...but can you edit them and save the changes from your mobile device? (It's going to depend on what software you have on the phone, but it's doubtful. From my BlackBerry, I could. From my Android, I can't).
3) When you do open something from Dropbox on my phone or tablet, it's actually downloading a copy onto that device. If I open that same document later, that's a second copy now downloaded. So, it's not a very clean setup.
As for the syncing problems, I have not had any problems. Not sure what to tell you there other than be sure you have the latest software update and search Apple forums for posts about problems syncing with Evernote.
I share your concern about having your data in a proprietary database. What if Evernote goes belly up? So many people depend on it, I think it's going to continue to be very stable. Never-the-less, you can export your stuff. For example, you can right-click a notebook and click "export." You can choose to export as html, for example.
I would encourage you to continue giving Evernote a try. At first, it may look like if you have Dropbox you don"t need Evernote, but having gone down that road, you really do need them both.