I still have a few weeks of money left on my claim and started a new job last week. Do I simply stop filling out the weekly log or do I fill it out answering that I did work that week and thus was not available to accept work and was not actively looking? I do not want to cheat the system but getting some of the remaining money allocated to me would help with my losses and if it is the policy to give out any percentage following new employment I would prefer that.

No, since you are no employed and no longer eligible for unemployment. Your unemployment stops literally the day you start yor new job. There is no money allocated for you. They don't set aside money to pay you all the way through if they have to.

you can't work and receive unemployment at the same time, that is fraud

Unemployment insurance is for the unemployed. If you are able to get a new job after 3 weeks of unemloyment, you are supposed to do that and stop collecting unemloyment insurance. You are supposed to be looking for a new job right away not waiting until the time limit is about to run out. You have already apparently abused the system and now you are suggesting committing fraud.