It's two parts:
1. Court:
Anyone can sue anyone, but it's an other question who will winn in court
In general verbal agreements are valid IF the law don't say written contracts are obligatoric (or there is a written contract, because such overrule verbal contracts, but that's isnät your case).
You can't be sure what a court will decide, but it seem your employer have broken the law already by not offer you a contract
(At least to kasambahays contracts HAVE TO be written AND REGISTRATED at baranggay office for such, and it would be odd if it isn't like that for other employee too.)
There are some exceptions because of you have worked so short time, and I know only partly which they are, so PERHAPS your employer have broken the law some concerning OBLIGATORIC employee benefits too. Some employers SYSTEMATLY try to avoid paying such benefits
and not offering you a contract is suspicious...
So at least you have some good arguments IF your employer will go to court.
2: Perhaps future problems:
Some employers demand to see documents from earlier employers, and some even phone them to hear their oppinion before deciding if employ you. So it's some risk your employer can give you problems with future employments.
So I don't know what's best for you in this case. But if your new employer offer you a good contract then he seem better, and you can get recomendations from him, which you can have use of in the future.