I would probably refuse the caution but only because having it on my record would affect my future employment due to the nature of my job.
There was no harassment when you went to her house - you say she was welcoming. Did she actually tell you to stay away?
How many abusive texts did you send and over how long? If it was over a short period you could rely on the defence that in the circumstances your actions were reasonable - for example, you had been in a long and stable relationship and were hurt and distressed when you found out she had been cheating. Have you contacted her in any way since your arrest?
Are you being honest when you say you didn't send threatening texts nor threaten her in any other way?
I would say refuse the caution if:
You have not contacted her since the arrest
You did not threaten her at all at any time
Your visits were only in the hope of persuading her to resume the relationship
Your texts were over a short time when you discovered the affair
Read these guidelines to help you decide what to do:
http://www.cps.gov.uk/legal/s_to_u/stalking_and_harassment/#a03a