My question involves a speeding ticket from the State of: Texas:
I was driving on the highway following the usual traffic speed. The speed limit is 60 but most traffickers travel anything between 60 -80. My car beong small and old I normally don't venture much high in speed.
As my luck run out one a certain unfortunate day I suddenly realised the cop car with his lights on just behind my car. I was around 60 that time. I realised probably its for me when I changed lane to right ( I was in 2nd lane from right) when he also followed me and I was confirmed he was after me. I stopped in the shoulder when he walked down to me he asked me if I was in hurry. I told him i was in hurry and I migt have travelled more than the speed limit.
He told me I was following too close.
Then he asked for all papers which i satisfied.
The ticket I received after that had these citations:

1>Speeding at 70 ( 70 written on top of 67)
2>"Following to close"

This is first time ticket. What are my options. I don't want the points to appear in my driving record.
Will this defence hold good?

a>First time Ticket
b>Any kind of deal where I plead "Guilty with explanation" to one and other dismissed. Pay fine and defense drive to take the points off the record for that one.
C>Does following too close result in points in driving history?

P.S. I am not sure about the "Following to close" and didn't ask much in that as a> this is the first time when cop was around me and b> I thought he would let me go off with a warning. But instead he gave the citations.

Thanks for any help in advance