Yes you can clean it out, but as always, there are some things to be aware of in the process.
- Use carb cleaner, not gasoline. Do this not only because it will work better, but for safety reasons as well. Brake cleaner would work well, too.
- When cleaning, you're only cleaning the pocket end (end away from the electrical connection). What you'd be doing is cleaning any varnish or carbon buildup around the valve itself. Do not submerge the IAC in any liquid.
- When cleaning, do NOT move the valve by hand. This will most likely destroy the motor (or gearing) which controls the valve.
- Be careful when removing the IAC so as not to ruin the gasket. If what I'm seeing is right, the gasket should be rubberized (not rubber, but the soft, pliable gasket type). It should be reusable, however, don't be surprised if you can't reuse it (ie: it might get damaged in the removal process). I realize you are on a budget, but don't skimp on getting a new one of the old one is trashed. You'll have bigger issues if you try to reinstall it and you have a constant air leak here.
- Not only clean the IAC, but in and around the intake manifold where the IAC resides.
- Clean the throttle body inlet and butterfly. You've gotten this far; go all in at this point.
Don't be surprised if, after cleaning, it still does the same thing. It could be the motor/gearing inside the thing is going bad and getting sloppy. It sounds as though it is actuating (or moving) the inlet valve very slowly, which is causing your issues. Cleaning this is a great first step. You could possibly get by with getting a used one from a junkyard, however, more than likely, anything you get there of this type will most likely either be in the same shape yours is, or very close to it. To me, for a part like this, you'd just be wasting money in the long run.