From that description, I tend to think they are unrelated, unless it's electrical. So you could start by checking the wiring to the fuel level sensor and the EVAP sensor and any associated fuses.
The fuel gage is inside the tank so a leak in the float will not leak to atmosphere and will cause it to sink to empty. So I would say this is most likely a wiring issue or failed sensor/sending unit. I think this can be expensive to replace on some cars because the fuel pump is on the same assembly, but maybe you can purchase it separately, you'll have to look that up for your car.
You can give the evap system a cursory inspection yourself. I'd start by checking the gas cap and making sure it's in good shape. Typically there are 2 or 3 metal lines that run the length of the car and rubber hoses on either end which are routed to the fuel tank at the rear and the fuel rail, carbon canister ect. at the front. If you can locate where these lines come up under the firewall and into the fuel tank, you can trace them and see the components of the system. You'll want to check the rubber lines and associated hardware, as that's where the leak would most likely be. If liquid gas is leaking, there could be a sticky residue near the leak. There will also be a vacuum line connected to the components of the system which you'll need to find as well since it could cause the problem if it is leaking or missing. All in all it is a lot to check and will take considerable time.
Personal non-expert advice, my brain says most likely electrical rather than an actual leak, with the most likely cause being a bad fuel tank sensor or bad wiring for the gas gage and a bad solenoid or sensor or missing vacuum line for the evap system. I don't think they're related.
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