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I might be missing something here. Is the doppler effect related to time dilation?
The actually true explanation is that special relativity only applies to reference frames without acceleration or nearby gravity.
So the fact that your clock runs slow as seen from earth and earth's clocks run slow from your POV doesn't cause a paradox.
There's no way to compare clocks at this time, since information can't travel faster than light.
When you decelerate/accelerate back towards earth, you leave the realm of symmetric time dilation and earth's clocks will appear to jump ahead as you switch reference frames, so it'll show more elapsed time when you're back. But I've yet to find an intuitive illustration for that.