You said that your COBRA was the same HDHP policy as before, so, yes, this coverage still counts as HDHP coverage. It does not matter if it was COBRA, what matters is the policy itself.
You would report that you had HDHP coverage from January to October. Note: it is the coverage that you have on the first day of the month that indicates the coverage for the month. If, by chance, you were still under the HDHP on November 1, 2018 because your new employer's coverage began later that month, then November also counts as an HDHP covered month.
As for the payments for the COBRA coverage itself, assuming that your employer did not subsidize the COBRA payments, if you have the money in the HSA, you can ask the HSA custodian to send you a reimbursement for the premiums - the COBRA premiums are a qualified medical expense (see page 9 at https://www.irs.gov/pub/irs-pdf/p969.pdf). If you no longer have enough funds in the HSA, you can deduct any COBRA health insurance premiums paid with after-tax dollars on Schedule A (Medical and Dental Expenses).
