Hand washing is good, and you are helping lessen your carbon footprint by doing so.
+1 for hand washing
Depending on how much you are washing dishes, how long the dishwasher lasts, and where your electricity comes from, the difference may be minimal if existent at all.
I am a dishwasher convert; I was adamant about handwashing for most of my life. "We used to wash by hands growing up so it's darn good enough for my kids" and that kind of thing. But a dishwasher really does use less water, even if you are washing your dishes really efficiently by hand. Partially filled basin with hot soapy water, rinse off suds by turning on the faucet. I can't realistically get under the 5ish gallons for a
full load that my dishwasher does. At first I only put place settings, but now I throw all of the pots and pans and casserole dishes and such in there. It cleans them all, first shot, with no prep other than scraping out the big bits.
I'm on electric grid heat so heating the water for hand washing uses
more carbon than the dishwasher heat (because I'm using more water). And water pumps for the cycle are all pretty efficient. So the real question is how long does the DW last, and is the embodied carbon in it more than what it saves over its lifespan? I don't know the answer; it might be quite large. But even if dish washing by hand uses a smaller carbon footprint overall, in the big picture it is quite small. I save much more carbon by doing other various things. Eating slightly less meat, for instance, will have a greater impact. Or wasting less food. Using cold cycle on your laundry. Using a broom instead of a vacuum. Taking the time that I now have from auto dish washing to hang dry clothes (I am trying to optimize my time, can you tell?). I could go on...
If you only have two place settings and gobs of time on your hands to wash dishes, then you are doing better than I am. I'm just trying to put things in perspective. DW is hardly the hill to die on (in my opinion) if you are trying to reduce carbon footprint.
* I would also like to point out that things will change once the isolation pattern lifts more. Depending on situation, OP might feel they like hand washing dishes right now because there's nothing else to do. But once you can go out more, the time spent washing dishes might be less appealing. I'm holding off on a lot of perceived lifestyle changes at the moment for that reason- I may not feel they are as needed once I'm out and about more.
But if you like washing dishes by hand, by no means take this as gospel, I'm just throwing opinions out there. I've done it most of my life and perhaps just trying to overcompensate for my conversion to dishwashers!