Yes, you absolutely should apply for citizenship. In some cases filing for N-400 forces decision on I-751, because you cannot get naturalized without conditions removed.
As long as you're LPR, you can stay and work in the US. Whether you have valid or expired GC doesn't change the fact you're LPR.
If you have unrestricted social security card and valid driver's license / state ID, you can show them to any employer to prove your eligibility to work.
If you have to travel internationally, you should carry expired GC and valid extension letter. If extension letter expired, you can get InfoPass appointment and 1 year stamp in your passport proving LPR status.
When you file for N-400 you'll get a letter similar to I-751 extension letter, but it won't be valid for travel or proving status as it only extends GC by 24 months.
Good luck!