Your employer needs to file

I-140 for yuu under the catagory of "Executive /Managerial positioin" (EB2). I assume you are at least a manager since you hold L-1A not L-1B. I-140 can get approved pretty quick depending on where you are in the US. One of our clients got I-140 approved within 2 weeks in Vermont Service Center.

Then you file I-485 together with EAD and AP. You still can travel under L-1A before you file I-485/EAD/AP. I suggest you to extend your L-1A (if has not reached the maxium of 7 years) until at least your I-140 gets approved. Once your I-140 approves, usually I-485 should not be a problem if there is no siginificant change of your company and your managerial position. I-485 can take years.

You can travel overseas even after you file I-485. You may travel still under L-1A visa or use AP.