Condolences on the passing of your mother.
Note that the date you get your checks is one-two months delay to the actual production of the gas. For example, a March check may be for January sales, so the prices will be for January.
This graph may help your understanding of the storage of natural gas throughout the year. Gas is stored ahead of the winter months and to some degree ahead of the hot AC months. Last year was an anomaly due to the vast exports of gas converted to LNG and sent to Europe to store up for their winter. Ukraine war and cut off of Russian oil and gas. Price is all about supply and demand, so gas prices were unusually high last year.
This graph shows the average price of Henry Hub spot gas prices. You did not mention your state, but prices vary from state to state. You can see the general uptick of gas prices over the last few years and a short term look at 2024. “Your results may vary”. Your checks are a combination of price for the product and the volume of product. Wells decline over time so you may get an uptick if a few more wells come online, but generally, with no new drilling and relatively stable prices, checks will go down due to declining volumes.
No, you cannot negotiate new leases with new percentages. You are held by the terms of the original lease and the wells that go with it.
The Mineral Help tab above is a useful place to start learning. Also the Oil and Gas Commission in your state probably has owner’s guides. Oklahoma does, so you can use it as a guide although laws differ from state to state.