It would be prudent to notify your homeowner's insurance carrier. However, there is some question of liability as to which dog dug under the fence; further, it seems that your dogs never left their property and the neighbor's dog again tried to enter your yard.
Sign nothing until you have heard from your insurance carrier. If you fear to bring a claim, find an attorney through the TX or your county bar association.
The foregoing is for general information purposes and does not establish an attorney-client relationship.
I agree with my colleague - don't let her bully you. Contact your insurance carrier and advise them IN WRITING that you do not want the claim settled.
If we do not have a signed fee agreement I am not your attorney and this is not legal advice.