We only do TV in the mornings so if she asks in the afternoon I always say no and she's fine with that. Best things to keep her busy are water and duplo figures. If I need to do anything in the kitchen I will get her on the toddler stool to 'wash up'. If I need to do anything upstairs I put a pool of water in the bathroom sink and she stands on her step to play in it. Sometimes I put a towel down on the floor with a plastic bowl of soapy water. I don't mind our rug getting a bit wet. She's also currently super into drawing and painting, probably because I've been doodling a lot. I've found that watercolours are actually better than kids paint because they won't spill and can't be grabbed at and smeared everywhere.
We reduced screen time gradually as I noticed she was starting to watch it quite a lot. Started off with no tv mornings then we said no to tv in the mornings and afternoon and she only got an hour around dinner time. We also noticed this reduced tantrums a lot that were tv related and she is okay now when we say no or turn it off. We live with family so she is usually entertained and goes into every room, we do drawing, she likes running around and follows me about, or I have to sit and play with her, something like blocks or she pretends to give me tea. But I would totally use screen time too if I had an another baby! Just do what you can as you need a rest too!
We just went cold Turkey on tv and the tantrums really reduced. Then we just introduced a very small amount at weekends and he doesn’t tantrum about it now. We do lots of music and nursery rhymes as an alternative to tv - and we have a Tonie box. General activities he loves at home atm - painting/ water magic paint, Playdoh, “drawing”, train track set. But - you have a few week old baby so I would totally give yourself a break about using “too much” tv! It’s probably a necessity sometimes. We’re about to have our second and I would totally use tv a bit more if it helps in the early days.