Based on
I have to sit there anyways.
I have limited extra non-work time, and am not churning out high quality applications at the rate I need to.
I am seriously considering quitting in the next month whether I have a job to go to or not.
No-one can see my screen.
I'd say discreetly work on your applications at your desk, but the priority would be doing any work your current job needs you to do first and with good quality. Just have a work-related tab open that you can quickly switch to if someone wanders over.
If my job was treating me like this (no active work, clearly not wanted or needed) I would feel no qualms about taking care of myself first. The only one who will take care of you, is you, don't expect a company to go out of their way to treat you well. But, this has a heavy caveat on my question below--
These sentences:
I am seriously considering quitting in the next month whether I have a job to go to or not.
If I do not get a job in the next month and a half I am stuck for another year.
seem to be at odds. Will you or wont you stay at your current company if nothing comes up? Saying you will quit regardless and then a few sentences later saying that you would be stuck at your current job doesn't compute. Which is it? That greatly affects the answer -- either you can afford to burn bridges accidentally or you can't.
If you are willing to burn bridges at your current company, I say stop caring so much and just look out for #1: yourself. And that means drawing a paycheck AND using your time efficiently, so hell yeah I would use my idle time to find better work.
If you are NOT willing/able to burn bridges, then you shouldn't do anything to put your current position in jeopardy. Only you can be the judge of that.
Edit: also, a month and a half isn't much time to do a job search, the process of searching and interviewing could potentially take that whole time. How is the job market in your field/area, are you confident you could quickly land a job with just a bit more time/effort, or are you down to the wire?