Here are my 2 cents. I'll preface this by saying there are always two sides to every story, we're only getting on side, so any judgement should take that into consideration. My first thought reading through this is that I think there are some additional details to be provided by the OP which would be helpful.
I was asked to sign/seal a set of drawings. I'm not a employee in a position that takes charge for work either.
I'm a bit confused by this. Do you ever seal any drawings? If you do then you must be in a position to take charge of the work. If you don't, then why do you have your PE? Or is it an issue of policy/practice at your company? To me anyone who has a PE should have the freedom and authority to lead a design and make their own decisions.
There are also situations where taking charge can take different forms. For example, what if the original PE left the company prior to sealing a design? The company would naturally look to another employee to take over that work and seal the design. For me, that would require enough time for me to thoroughly review all the work, assumptions, correspondence, etc. done up to that point and have the freedom and authority to make changes as needed before putting my seal on it.
First, I'm not a subject matter expert of the type of design and calculations to be certified.
How different was the work from what you currently have knowledge in? I wouldn't expect an electrical engineer to seal mechanical drawings, etc. Neither would I expect someone who does HVAC design to seal part design drawings. Those are very different and would consider that a violation of state law. However, if it's closely related and applies the same general principles with maybe a different application, then as an employer I would expect someone who is competent in the one area, could do some brief study and discussion with peers who have more experience in another area to feel comfortable. I see it essentially as the learning and growing process in any career. That being said, if you simply aren't comfortable then no one can compel you to seal against your will. That would be part of a negotiation with your employer because they may feel there is enough correlation that it shouldn't be an issue. Patience and humility are key here.
Secondly, I don't have the license in that state.
There are situations where you don't have to be licensed in the state to seal. For example, many federal projects don't require it if the project is on federally owned property. In that scenario, the state has no jurisdiction and the federal govt will allow home state seals to be used. Thinking in particular construction design drawings for a building. Otherwise, you are correct and that should essentially end the discussion. Most states also require you to be licensed in that state from the very beginning of the project. So it's not like you could go get licensed midway or at the end and then seal it. Check with your state.
And third, told to 'relax' because 'we have insurance'.
I suppose I'd like to hear the context in which this statement was uttered. If they're using it as an excuse to break the law, then absolutely do not do that. I'm struggling to think of a legitimate reason for them to mention this. Sounds like they are trying to coerce you into sealing a design you don't want to seal, and as mentioned above, you have legal standing to reject that if you're not comfortable. Just make sure you're being reasonable with what is being asked.
I was further pressed that this 'had to be done' and to stop 'looking for attention'. In my opinion, this should be a question of why someone in a position that could take charge of the work doesn't have a PE and why this responsibility gets kicked down the road. Regardless, there's no way to force someone to sign and seal drawings.
Again, would like to hear the context and both sides of this exchange. Generally I do agree though that whoever the project manager was for that project should have nailed down the sealing engineer from the very beginning and had that person involved from the start. That's usually required by state law. However, there are situations where factors outside their control force them to look for another PE and there are ways to navigate through that. Occasionally people will also mess up and not do the right thing, in that case it goes back to having the ability to review all work and make changes, etc.
I offered a resignation if I were to be further pressed. Ultimately, it's my reputation and livelihood at stake. This has received mixed feedback that I blew everything out of proportion and shouldn't have offered a resignation but instead used this as an opportunity to negotiate a promotion/raise/etc. Some have said I should have resigned on a spot and started searching for another job.
If what you're saying in the rest of your post is true and accurately reflects the situation this was probably not a bad move. Before making a change though I would carefully consider the nature of their disagreement. Are they simply ignorant of the requirements and that explains their frustration, or is it that they know but don't care? If they know and don't care that is a deal breaker for me and I'd be looking for a job ASAP. If they're just ignorant or misapplying the law then there is room there to work with them and see where it goes. Maybe brush up on your resume in the meantime just in case.
Hope this helps.