Update: So I checked Mr. Bean's bum today and it was looking more noticeably wetter/dirtier. I think he was still normal, his coat still looked good, he had run on his wheel last night and he ate his food and even drank water before we took him to the vet.
While the vet checked up Mr. Bean, she noted that he was still bright and alert, hydrated, and wasn't hunched or feeling pain in his stomach. She even checked his mouth and said it was good. I believe he also weighed 126 g on the scale. The vet concluded that Mr. Bean was probably just having straight up bad diarrhea from the stress of being in a new home/environment, and prescribed an antibiotic to treat it. She said that even if he did have wet tail, this would technically be the early stages and that the antibiotic would treat it if that's what he has. Although, she did also mention that he is kinda too old to have it. She said his chances look good since he is not lethargic and we caught on early, but did say that it's not 100% guaranteed. The antibiotic treatment is a 2x a day treatment through a dropper and if he doesn't get better in a week (5-7 days) then to call the vets again.
I'm relieved to hear that the situation isn't as bad as I thought, but I'm still fearful that the medicine won't work and that Mr. Bean won't get better. But I really hope he does <3