Are the spots visible only on solid-colored objects (sky, wall) or do they ruin almost every frame? This matters because if they're only noticeable under specific conditions, you can get away with software correction in post-processing.
If you do decide to clean it yourself - yeah, specialized kits work, but there's genuinely a high risk. The difference is you need to be super careful and first make absolutely sure the dirt is actually on the sensor, not on the mirror or protective glass in front of it (you check this by looking at photos shot in live view mode, when the mirror is up). A blower bulb is the safest move; a brush is already riskier. Personally, I'd seriously consider professional cleaning in your situation, because even the slightest wrong touch to the sensor can cause artifacts on your images afterward.