Heavy corrosion is not an indicator of a bad battery (unless your battery post is so heavily corroded that it is almosted corroded into just below the battery case!). You need to keep some vasoline on the posts and terminals to seal it away from the air to prevent this.

If you are getting better than 13.5 VDC at the battery terminals when the engine is running, then chances are pretty good that your charging system is ok.

Take your terminal connectors off. Clean them good and also clean the battery posts. Put the connectors back on and make sure they are tight.

My bet is your battery is bad. If you don't have the tools to full load test it, you can take it to an AutoZone (or similar) and most will test them free for you. That is the only way to tell for sure.