You're totally right, it's a dead-end job similar to a paralegal. Both tend to be nestled underneath general counsel and it's not a highly valued corporate function with an upward career path. So if you want to move upward, you'll need to lateral first.
One idea is to work on large proposals and to bridge over to business development. It will take awhile, but if you're high impact on the proposal team then the BD people will want you. Then you can become bonus-based and there's definitely an upward path on the sales side.
The other idea is along the lines of what
@tyler2you mentioned. Complex regs and litigation in the gov space means contracts experts are more highly valued and there's a security clearance bump. You could try the GS route or government contractor route -- probably just try both.
Lastly I'm sure you do a lot of things that aren't strictly contracts administration. Maybe strategy? Some finance? If you peruse job listings, you might start to see opportunities where you're a stretch fit. You can either apply for those or see how you can do more of that function in your current role to qualify.
With this record hot job market though, I'd make a plan where you're moving roles sooner rather than later. Hiring managers are willing to look at stretch candidates. YMMV
PM me if you want to talk specifics. Good luck!!