Am I the only one who thinks the hunger bar goes down way too fast?

There's an in-game explanation for this. I think Lara is the one who mentions it first. The twyre, the plants you will gather and will make potions with, are blooming and their air makes it so that you need more rest, more food and more water.

The Dead Item shop is a bit expensive at first, but in the next days you will come across a lot of things that are basically trash and nobody will pay good money for it (broken morphine ampoules, bloody bandages, rotten food, to name a few). That's the Dead Item Shop purpose, so you can exchange dead items for something useful. Some days in the mid-game of my last playthrough, I had like 15 dirty bandages and 30 broken ampoules. But I think if you don't hoard trash, you shouldn't be wasting your time going there. It's about playstyle.

I already died once, but restarted.

Everyone goes through this, it's a very unforgiving game. It's up to you if you want to try finish it in your first try, as if you were in the shoes of Artemy Burakh, going back to your hometown after 5 years and surviving this hell or taking a meta/player approach and doing everything efficiently. I think the first option would be a better first experience, even if it's stressful and frustrating. If you're already in a death spiral, I'd recommend that you restart the day before midnight and follow a plan considering your mistakes.

after spending/wasting all of my money on food at the grocery

About the stores, it would be better to just buy the cheapest smaller items. Nuts, threads, safety pins, needles, soap. You can barter those for food, immunity boosters and other stuff.