A bit about my background: BSc. Comp Sci, MBA, MSc Software Engineering. 18 years in software industry doing Tech Support, QA, Tech Writing, Software Development, Engineering Management, with 10 years doing the latter two at Google before retiring at age 38. During my time at Google I did somewhere around 300 interviews for software engineering candidates.
Computer Science is an extremely broad field, and the scope of possibilities expands significantly within the even broader tech industry. You're currently doing security systems administration, and it doesn't sound like you're doing any programming in that gig. So my first question is what type of work, specifically, do you want to be doing in industry? Admin? Security research? Database? Software engineering (this in itself is rather broad, with many specialized areas: fullstack, systems, infrastructure, architecture, ...), AI/ML, ...?
I'm going to assume, based on your background and other things you mentioned, that you're interested in going in the general direction of software engineering. You have the CS degree, which is a great start, but this really is just a start. It will help you get your foot in the door, but it is typically not sufficient on its own. The CS degree gives you an important foundation, and you should keep up on what you've learned so that you can competently analyze the runtime complexity of an algorithm, be very comfortable with the tradeoffs of all the main data structures and algorithms, and be able to reason about different space and time tradeoffs. But in addition to this you need to get lots of programming experience. There's a world of difference between working on a 1000 line undergraduate programming assignment vs. a commercial product with hundreds of thousands of lines of code across a team of people.
As you pointed out there's a bootstrapping problem where it's difficult to get the experience needed to be a competent developer when you don't have the experience to get into your first job. All I can say to that is that it's difficult but possible, and there are many different paths. Some people get involved in open source as a way to gain experience. Others get in at some desperate place that cannot afford to be picky, perhaps a startup that is having difficulty attracting people because it's not an exciting product. Sometimes it possible to start somewhere in a different but related area with the possibility to move into a software engineering role. If all else fails, you can (and should anyway) always be coding on your own time. Think up a few fun projects, ideas, or just fun coding problems, and write lots of code on your own. Code, code, and code some more. Learn different languages and frameworks which will make it possible for you to write code efficiently and confidently in an interview, which will go a long way to helping you get established in the field.
One final observation: IMO Universities are not a great place for career growth - they are too large and lumbering, same goes for government jobs. A temp University job is great while you're in school, but at this point I would start pounding the pavement looking for something in private industry.
I will also second what other's on this thread have said about job requirements: If you have even 25% of what they are listing then go ahead and apply. It doesn't hurt to try, and any interviewing experience you can get is worth it.