How can I approach looking for an FTE role?

For the manager conversation:

• It is really hard to bring this up without sounding accusatory. Unless you have a good amount of conviction in your manager being good and liking you, I would err on the side of not doing this conversation at all.
• If you are going to bring it up, do it extremely delicately (feel free to share some theoretical dialogue, so the community here can share feedback). I would say something more innocent like, "Just noticed that I hired several full-time engineers like me: Does this mean that there's headcount open for FTE now?"
• There's a clear power-dynamic between you and your manager. The failure mode is envisioning some magic scenario where you successfully call them out and they give you a full-time position from that: That's not very likely to happen.

For contract vs. full-time:

• I think the main benefit of contracting is the flexibility. Hours can be lower, and you're not held to the full performance review process of the company. At Amazon, I imagine performance review is rough.
• On top of the pay/benefits, I think the main thing about being full-time is that the company is probably going to invest more into you. This means better projects, more face time with your manager/TL, etc.