We moved in to take care of mom, I got sick also and depend on government help to live. Do I put us on one household?

For the Food Stamps you can probably be 2 households. Generally if all the food is purchased and prepared together there is one household and all the income is counted and size of the entire group is considered. However, if Mom needs assistance to do all that because of her health, she could be a separate household. In that case you need an application for her Food Stamps and an application for the rest of the family. You list everyone in the household on both applications. The difference is that on Mom's app she is the only one applying for assistance and on the other app, the rest of the household is applying, not including her. For the income and expenses part just put what is appropriate for each group. Mom's app only has her income and expenses, the other app has the income and expenses for the rest of the people. It is not uncommon to have more than one "household" under one roof. As to Medicaid, is Mom the only one applying for assistance? Usually the application for aid has several boxes to check...so check off both the boxes for Mom's FS and Medicaid. Do the same on the other app if the rest of the household is also applying for Medicaid. I do not think all should be on one application if Mom can qualify on her own and that should be to your financial advantage. You should also look into resources for assistance for rent and utilities. There may also be discounted assistance for transportation. One other possible benefit, if you do qualify for FS, also ask about commodities distributions. These are actual food items provided by the federal government, usually distributed by charities like Catholic Charities, on some sort if schedule. Think of 5 pound blocks of cheese, etc. distributed quarterly. If you get FS you also qualify for commodities.

Medicaid looks at household income and assets, so list everyone at the same address and all the assets and income, what you pay in rent/mortgage.
Don't try to hide assets, that would be big trouble.