Trying to resolve problems like this with only generic input is difficult. It is much better if you were to provide screen shots of your particular documents or other numeric data. Under the circumstances, here is the best advice I can offer.
Within Photoshop the "dpi" setting (should be ppi or pixels per inch) is only meta data that comes into play when you try and produce a print of some sort. Since you are intending to print, this setting matters, but the pixel count in the image also matters. What you should do is first determine your print size, lets say 3 inches by 2 inches as an example. At 300 pixels per inch you want to create your Photoshop document to be 900 pixels by 300 pixels.
Create your logo artwork and text within Photshop. Ideally, both would be vector entities, that is, made up of shape layers and normal text layers. The text should not be rasterized unless there is some compelling need. If you do something like warping text it will remain vectorized in Photoshop, but will be rasterized when digested by Illustrator. You can then save your psd file and it should print fine at its intended size, from Illustrator. When you open the psd file into Illustrator choose the "Convert Layers to Objects" option otherwise it will enter as a single layer image (rasterized). I suggest you make test prints with your desktop printer using both Photoshop & Illustrator
to monitor the results of using Illustrator as a print engine. I would typically do the opposite and use Photoshop as a print engine for Illustrator documents.
Paulo
