Vic,
I sometimes wish I had your problem. Here are the selling points and vectors.
Know your customers! The app teams are usually the drivers and if you show them how much you can do for/with them they will likely see the light.
Single threat surface that can be managed universally with updates and policies that can be quickly defined and tuned to fix or avoid costly security issues.
How many times have they had an SSL cert expire without knowing ahead of time.
Easy to do with LTM and braid dead easy with LTM and EM.
Don't forget centralized authentication to all applications and the best thing is the end-user can't even reach the server until they are authorized so you again are more secure.
Almost every outage that happens or delay in service I can show where it would have worked better or not even have happened with the proper LTM configuration in front of the application.
Good health monitors that actually detect the application's worthiness to handle traffic are fundamental. With these and the LTM SNMP system you have a very close to real time picture of what's happened/or going to.
Again, grass roots your passion and knowledge into the groups/people that can really see the benefits. Have more casual conversations with them and tease out their worst problems. Come back later and mention you have a solution or part of the solution with the technology you manage.
Good luck.