Yikes. That's tough. I've managed people for a decade and can't imagine not interacting with them regularly. I'm in Massachusetts and have had people work for me in California and I still met with them (via phone) at least once or twice a month to check in, discuss progress, and give them an opportunity to interface with their manager. I can understand why you're anxious. Even without the distance these types of conversations can be difficult. But the topics you want to review really should be discussed via conf call if at all possible. If you were local you'd broach the topic in person so I think a phone call is a reasonable alternative given the situation.
You know your company and situation best and should proceed however you feel comfortable. I will give you my suggestion just for consideration.
If I were in your situation, I would email my manager something to this effect:
Hi [Manager],
I know your schedule is busy but would it be possible to schedule a quick conference call to chat? I've been thinking about how to increase my involvement in the company and would very much appreciate your input.
Thanks,
[Name]
The message is short, gets to the point without being confrontational, and is posed in a way that you're soliciting his insight. If he comes back and says "yes" then you're fine to proceed. If he says no then at least you tried and you can follow up via email.
If/when you talk with him I would try to stick to the key topics and leave emotion and personal info (e.g. insurance) out of the conversation unless he asks direct questions. If your objective is to obtain a full-time position then you could start by expressing that you've enjoyed working on the team and was wondering if there is an opportunity for you to transition into a full-time role. It's a simple question and should be asked nicely. You may not love the answer but at least you'll have it. If he says "no", you could follow up with "Okay, I understand and appreciate your letting me know. Do you have any suggestions on how I might be able to expand my role in the company? I'd really appreciate your insight."
That's just my thinking
I have some thoughts to share after reading the earlier posts:
-Is there a reason your team lead hasn't addressed this issue with your manager? I'm not clear what the hierarchy is. I just want to ensure that it's appropriate that you address this with your manager directly.
-Part of the reason people hire part-time employees is because the company doesn't have to pay benefits. It's not a shortcoming with your individual situation; it's the norm and therefore shouldn't be referred to as a problem (I'm not suggesting that you are doing that).
-If your company publishes job postings have you looked at them to see if there is a full-time position you could apply for?
-Re raises: In every company I've been with they're only given during annual reviews unless the manager takes the initiative to restructure/redefine a position (which is not common in my experience). I'm not sure how long you've been in your position and if you've had an annual review but if you're asking for a salary adjustment mid-cycle I don't know how likely you are to get it. However, as a part-time employee if you're paid hourly it may be reasonable to bring up the rate if you're not stuck on a regular adjustment cycle...