Last Friday, I noticed one of my cat's claws lookturned black.

Or "Looked dark" compared to the rest of her claws.

I thought I shouldneeded to take her to the vet right away, but the vet was supposed to be closed until next Tuesday.

Since the context is urgent, using "need" works best for this phrase and since you're talking about the past, it would become "needed" as past-tense.

I called them to ask if they would see my cat on Saturday because I thought it might be an emergency.

Another you can say, "I thought it might be urgent."

However, they said "Sorry we can't see your cat because the vet is away" andso I booked my appointment on Tuesday.

I was worried about leaving her claw leavingalone (or "the way it is") for four days.

OPTION A (In MLA Format): My Cat Has a Problem on hHer Claw¶
OPTION B (In APA Format): My cat has a claw problem

Last Friday, I noticed one of my cat claws looked blackdarkly sallow.

I thought I should take her to the vet right away, but the vet was supposedchedule to be closed until next Tuesday.

I called them if they would be willing see my cat on Saturday because I thought it mightseemed to be an emergency.

However, they saidreplied, "Sorry we can't see your cat because the vet is away" and I booked my appointment on Tuesday.

MY GUESS: I was worried about her claw leaving for four daysa possible claw infection.

NOTE: The phrase "for four days" can be deleted - readers can infer that information.

ALSO POSSIBLE: Today, I finally took my cat to the vet and found out her claw washad become infected.

She wore a cone around her neck to prevent her from licking her claw.