Yes, I think you should keep trying... if this is what you think you actually want =)
About the side projects... At least in my team, one of the reasons we ask for side projects, is because is a way to know that you are able to build something... and that you have at least deal with the things required to ship that something...
One advice that I can give you (maybe because is my experience) is that is easier, to get accepted in companies that have, apprenticeship programs. Those companies are a good way to start... and maybe continue there =)
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