You can switch jobs with EAD in hand.
Once you receive GC you can:
1) Go to SSA office, file SS-5 form and replace SS card to remove restrictions from it (e.g. wording "Valid for work only with DHS authorization")
2) Use Drivers License / State ID in conjunction with SS card for I-9 verification moving forward.
This way you don't need to worry about showing GC and if it's conditional, extension letters and all of that stuff. Some HRs get confused about GC expiration, extension letters etc. It's far easier using documentation a regular American uses. Obviously, make sure to use term Lawful Permanent Resident and never say you're a US citizen until you become one.
It's just me, but I'd go as far as obstaining from using words "Green card holder", "I have a Green card" because that's what triggers HRs / employers to ask for GC as proof, which has multiple issues as outlined above. I always say "I'm a lawful permanent resident and don't require work authorization" whenever asked about status.