"Im feeling a little under-appreciated. I fear that the only way to gain my appreciation back, is to do the impossible... and manage everything efficiently and flawlessly like I was before I had an endless task-list of projects + support."
My experience tells me that it is more like the opposite.. if things run smoothly, people will just asume that your job is easy.
For a week or two, document all of your activities. Also document tasks that are waiting for you to have "a little bit of time".
Then go with those documents to your manager and tell him with that data the nature of the situation, and also your sinking morale.
It is your manager task to solve the issue. If your manager has any doubt about your capabilities, just candidly ask him (or anyone he designates to) to sit with you for a couple of weeks so he can make "suggestions" to improve your efficiency.
After that point has been solved to his satisfaction, I would not make any proposal. He now knows the issue and he is the person in charge to solve it, let it be by hiring more people, increasing your contract hours, whatever.
Do not be shy about it. The worst your boss may do to you is firing you, and from the tone of your post, it seems that if things do not change you will not be able to last long in your job without severe issues of stress/depression. In case your boss does not want to listen to you, it may be better to you to leave now as a sane person than in six months because you have suffered a mental breakdown.
Edit: to clarify about the "I would not make any proposal" point, and answer to Jan Doggen's comment.
Going to your boss and stating "if you make me a full time employee all those issues will dissapear" can have several outcomes:
you get the fulltime contract and enough time to handle your workload so your organization performs efficiently. Nothing to say against this.
your boss may suspect that you are making up or exagerating the issues in order to get that position. That could make it harder to explain the situation to him. It will depend a lot on your boss' experience, his human quality and/or his preconceived ideas about you, so I cannot tell how likely that would be.
(my main objection) you get the fulltime position, but you are still overworked (because it is not enough or even worse, since you are now "full time", your boss decides to add some other responsibilities to you). Now, you did promise to solve these issues if you got the contract, so so you have lost a lot of leverage against your boss. That would make the promotion a curse in disguise.
Of course, when exposing issues to your job, you may say that a full time position would be a nice way of rewarding the energy that you have put on for the previous year. Unless your boss is retarded, he will see that one of the easier and faster solutions is that you work more hours.
If the possibility is stated, you could make some reassuring statements in the line of "I will keep working with the same intensity" or even (if you are pretty sure of your estimates), something like "If the workload keeps being the same, I expect to be able to improve the issue significantly". Note that here you promise to do your work well, not to solve all the issues that the company might have (because those depend of lots of factors that you do not control).