That's a lot of dust.
The blobs are so extensive and ugly that any software is going to have a very hard time making a good image from these.
Thje old saying is that "You can't make a silk purse from a sow's ear"and, unless these are vastly worse than actual, this seems to have got to sows-ear stage. A serious attempt at finding a low cost solution seems in order.
So - the following is about reducing the effect - not on post processing:
You will get more spot visibility on higher apertures.
Anything you can do that will keep aperture large (= small f numbers) will improve the result.
This is because, if the dust is close to the sensor it will be actively de-focused and large apertures increase this effect.
The C1035 has variable ISO over a wide range.
Try taking photos at the lowest acceptable ISO and see if it makes a visible difference.
Low ISO means the camera needs more light and will a lower aperture and shutter speed.
Downside: In lower light conditions this will lead to low shutter speeds and you may get blur you otherwise would not have. Large apertures give reduced depth of field so focusing is more critical, but in many cases with a small sensor that will not be too much of a problem. Try it and see.
You will pass this way but once, or maybe a few times in your case, you being more traveled than most. Years from now your photos may provide the major link to memories of your past. Spending the minimum necessary that is required to allow the photos to be a bearable record of events is an investment in your future. What country are you in at present? Depending on where you are, you may be able to get a good used camera that does a better job overall than the dusty Benq for perhaps $50. The Benq is 10 mp. A good 4 mp camera will provide a very adequate record in most cases. Investing in another camera may be a good idea.
You MAY be surprised how little a sympathetic repairer would charge to clean your sensor. If you can access the surface then a blower may help. Gentle brushing with clean lens tissue may also help. I would be personally prepared to attempt using eg soft toilet tissue a few turns inside the roll to ensure it is dust free. This is far far from lint free but stil usable en extremis. People sell replacement LCDs for the C1035 on the internet. They may be prepared to provide dismantling instructions for free that a competent person can follow. An evenings careful playing in a hostel dining room may be all that is required.
All care & no responsibility suggestion: I personally would attempt using a vacuum cleaner and great care to suck air through the camera - with due care to prevent full suction being exerted. With proper care this carries minimal risk of damage and a moderate chance of success. I'd open the lens fully out, remove the battery and apply careful sucction around lens tip and battery box. Fingers over hose end to stop it latching on and care so the camera never gets pulled hard against hose.