Answer:
This question isn't actually specific to open source licenses — it's really just about how to apply some particular copyright license (whether open source or not) to your software. Please note that the OSI is not a legal services organization and does not provide legal advice. However, many licenses come with instructions on how to apply them: for example, see the section "APPENDIX: How to apply the Apache License to your work" in the Apache-2.0 license, or the section "How to Apply These Terms to Your New Programs" in the GPL-3.0 license. If the license you want to apply has such instructions, just follow them. If it does not, then look at the previous two examples (or at other licenses that contain similar instructions) and follow a similar recipe, adjusting for the license you're actually using of course. The Software Freedom Law Center also maintains a guide on managing copyright information within open source projects. Finally, this guide may also help, though please note again that neither it nor this FAQ item constitutes legal advice.
Note that releasing software under an open source license does not involve contacting the OSI, signing up to some process, or handing a copy of your software to the OSI or any other organization for evaluation. You just publish the software with an OSI-approved open source license attached, in the manner described above — that's all you need to do.