Attempting to rehome some strays

These two strays were dumped around my house and have just been hanging around for the last few days. They’re very friendly sweethearts that come up to you when you call them and are pretty chill. My mum has absolutely fallen in love with the black one and wants to keep him but unfortunately the blond one isn’t getting along that well with my dog I already have and I wouldn’t want an unnecessary fight to break out.

So I was wondering what the best option would be here? All the shelters Ive called in the area will not take them as theyre full and I was warned about putting either of them up on facebook.

I feel a bit bad thinking about separating them should we keep them together? And Ill also need to take the black one to the vet anyway for general checkups. Should I also go ahead and set up an appointment for the blond one to get her checked up before continuing to find her a home? Or will the vet have better options on what to do with them?