No, I don't think it should be diverted.
Foreign Aid is drop in the ocean of the social care budget (which is gigantic and growing) - the money that is being spent on aid will have relatively little impact if it is diverted there, whereas it can have much more significant impact where it is, because projects in less developed countries are cheaper than year-on-year care for the elderly in a developed one. So Foreign Aid is a small amount of money that does a lot where it is, but wouldn't do much if it was moved.
The only way to solve the crisis in social care is to raise more money and to invest it properly. Either through an increase in corporation tax, income tax, or borrowing. Each has its own drawbacks. But we are not going to get the amounts needed by cutting other departments - it just isn't enough.