Is is necessary to probate my deceased mother's real property or can I transfer it to me under Texas probate laws

Since there is a will, you need to probate it in order to transfer title. Until it is probated, there is no individual who can legally act on behalf of your mother. Also, the will must be probated within 4 years of your mother passing. If all debts of the estate have been satisfied, then you can use a more simplified probate procedure.

While I am not a lawyer in Texas, I do not feel that it would be illegal to leave the property in your mothers name, but it will create a title nightmare if you ever want to sell or finance the property, or leave it to your children. The clean way to handle this transaction would be for the Executor of the Estate and your brother and sister to sign a Deed granting you title, which deed should be recorded in the clerk's office. Not changing the title now leaves too many open questions, and too much of an opportunity for something to go wrong. I suggest that you contact a local attorney or title agency to take care of this for you.

In California it is not a criminal offense to fail to probate. I assume that same holds true in Texas.
With that said, you are doing a disservice to yourself and your heirs by not transferring title.
By transferring title you will be able to finance the property and leave it as part of your estate to
your heirs.