Ex ante: There seems to be a lot of confusion on what is regarded as testing what is not. Sure, every developer needs to test his code as he creates it, he/she needs to verify it works. She/he can't hand it to a tester before he/she thinks it's done and good enough. But developers don't see everything. They might not recognize bugs. These bugs can only be found later in the development cycle when thorough testing is conducted. The question is whether developers should conduct that kind of testing or not, and in my humble opinion this needs to be looked at from a project manager's point of view:
Developers can be testers, but they shouldn't be testers. Developers tend to unintentionally/unconciously avoid to use the application in a way that might break it. That's because they wrote it and mostly test it in the way it should be used.
A good tester on the other hand, tries to to torture the application. His/her primary intention is to break it. They often use the application in ways developers wouldn't have imagined. They're closer to the users than the developer and often times have a different approach to test a workflow.
Also, using developers as testers increases development costs and does not benefit the quality of the product as much as having a dedicated tester. I wouldn't let developers cross-test their works when I can have it done better by a tester for cheap. Only if the feedback loop between developers and testers became too expensive, I'd have developers crosstest each other's code, but in my experience that is rarely the case and it highly depends on the process.
That does not mean a developer should be sloppy and leave everything to the tester. The software should be backed up by unit tests and technical errors should be reduced to a minimum before handing the software to the tester. Still, sometimes you have fix here, break there problems or other bugs that no developer could forsee, that's ok. Also, integration testing should be done mostly by the developers. The tester's main objective is to verify that the requirements are met.
In such a small team (and also depending on the size of the application), I can also see the tester in a hybrid role, writing unit tests and UI tests. You should definitely hire one.
But more important than the tester are regular freezes/branches. Don't present anything that hasn't been properly tested. When you've added a feature or changed something, everything surrounding it has to be verified again. You'll only get a bad reputation if your company doesn't. Don't release something unstable. When the customer wants to have the software by a certain date, then stop developing early enough and test it properly, so you have enough time for bug fixes. Often it's better to decline last-minute feature-requests than to implement them poorly or release without proper testing.