You can keep your background in Photoshop and continue to use Photoshop to manage the calibration, colors, resolution and effects as you like to do.
And for your texts, you can save them in a new layer file in Photoshop (a .psd), and then open this file in Illustrator.
Here is the way to do it
Once you have your text in vector (converted from your Photoshop layer file), you can simply add your background to it and use Illustrator as your new publishing software. You should now type all your texts and draw the sharp graphic in Illustrator, and not use Photoshop anymore for this.
Yes, ideally you should use Indesign especially if you need to do multiple pages layout. But Illustrator can be used for single page layouts such as postcards or business cards.
When you'll prepare your files for print-ready use, you can select all your text and do a "create outline" (menu "type") and it will make that text non-editable. Then you can save that file to PDF with the highest quality possible. (Some tricks to compress a PDF to lower size here).
if you need to create a multiple page document but still prefer to stick to Illustrator, it's always possible to merge all your PDF files in ONE single PDF file. This is how it's done if you can use Adobe Acrobat Pro that comes with the Creative Suite/Cloud from Adobe.
You could always keep working with Photoshop but... You really gain more in simply learning Illustrator and even better, Indesign. In Indesign, you can easily combine together your vector graphics from Illustrator and you pictures from Photoshop; and the best thing is InDesign creates vector texts that are always perfect for printing and is also created to handle files to the max performance of your computer. You will find this interesting if you ever need to work on a catalog for example! You can also use Master-Pages with Indesign and it saves a lot of work. Indesign is really better than Illustrator for publishing and productivity.
Note: Exporting a PDF from Photoshop WITHOUT flattening your layers should keep your text layers in vectors. Use the "save as", no need to "print" the PDF.