"Do I need to submit Apply Texas essays B or C?"
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No. Under no circumstances do any universities or scholarships require additional long Apply Texas essays beyond Essay A.
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Texas Public University Essay Requirements
UT-Austin and Texas A&M are clear that they only accept Essay A and will not allow you to submit Essays B (discuss and identity, interest, or talent) or C (you have a ticket in your hand).
UT-Dallas allows students the option to submit one of the Apply Texas essays, but in practice everyone submits Essay A so they don’t need to write a second essay unnecessarily. There are no bonus points for submitting additional essays.
UT-Dallas, UT-Austin, and Texas A&M are the only Texas public universities that even require any essays for regular admission. All of the remaining Texas public universities you can apply and gain admission based on your class rank and ACT/SAT scores alone.
Texas Private University Essay Requirements
Many Texas private universities are on the Common Application or Apply Texas. Some of them require an admissions essay, and to my knowledge, none require more than one. Austin College, Trinity University, St. Edwards, Baylor, and Southern Methodist University (SMU) requires one essay of your choice from either the Common Application or Apply Texas.
Rice University is only on the Common Application and requires a few supplemental essays in addition to the main essay 250-650 word essay.
Texas Christian University (TCU) has a few specific prompts not on Common Application or Apply Texas where you must choose and respond to one, but it would be rather easy to modify a UT Austin Short Answer or Essay A Tell Us Your Story to answer them.
Note that some specific majors, honors programs, and scholarships for public and private Texas universities may require additional supplements or essays.