You ONLY enter the W2 box 12 code W amount ... later when it asks for contributions to the HSA that you made Outside of payroll you should not have entered anything from the box 14 amount ... return to that section and remove the erroneous entry. 

The HSA is handled in 3 parts in the TT program :

First the contribution:

https://ttlc.intuit.com/replies/4557768

https://ttlc.intuit.com/replies/4785646

Next the limitations screen to confirm you are eligible to make the contributions:

Until you complete the HSA portion of the TurboTax interview to establish your eligibility for an HSA contribution, TurboTax will treat the amount entered on the W-2 form as an excess HSA contribution.

https://ttlc.intuit.com/replies/4788059

And lastly any distribution:

https://ttlc.intuit.com/replies/4787864

Perhaps I'm not explaining things clearly (or still not getting what's wrong)

My W2 has two entries in Section 12  

C- $435.60 (group term life insurance)

DD- Cost of Employer sponsored health care - $4326.21

There is NO "W" entry for HSA contributions.

In the HSA section- my personal contribution of $6637 is entered  (of which 674 is excess)

My former employer confirmed that Box 14 on W2 in the amount of $2523.96 included all of my pretax contributions-$1261.44 for health, dental, vision) plus MY payroll HSA contribution of $1262.52 

The 5498 HSA form correctly states the combined employer/employee contribution is $1937 (1262.52 for me, 675 from my employer).

The HSA Summary page shows:

Deduction.  5963

Total distributions. 2541

Tax Free employer contributions. $1937.

If I remove the entry from "Employer and payroll contributions not reported in Box 12 of your W-2", then my federal tax goes down by $202 and CA tax goes down by $180.    If I reenter the $1937 in Box 12 W-  taxes go back to where they were when amount was entered in the HSA section.

I've checked the answers multiple times for myself and husband.  

1) No we did not take Medicare,

2)  Yes we had a family HDHP for entire 2018

3) Entered HSA value as of 12/31/2018 (including 2019 personal contribution)

Next screen shows Summary as stated in previous reply.

These are the instructions I followed to enter the $1937 in the HSA Medical section.

But what if I made an HSA contribution that wasn't reported on my W-2?

We'll ask about this uncommon situation when you step through the Medical section. Look for the questions Were HSA contributions made in 2018? or Do you need to make any adjustments to your HSA contribution amount?When you answer Yes, we'll open a screen where you can enter your contribution.

Your entry method is correct [although the amounts you've entered might be incorrect; see my later comment below].

The increase in CA tax makes sense since CA does not conform to the federal deduction for HSA contributions.

[Disregard this paragraph.  The explanation is in my comment later comment below]  The increase in federal taxes is a bit puzzling.  Perhaps it's the result of a side effect of the reduction of your AGI, maybe a reduction in Earned Income Credit.  Compare your Form 1040 with and without the HSA contribution entry to see where the increase in tax is coming from.  Check to see that Form 8889 shows $1,937 on line 13 when the HSA contribution entry is present.

I also just confirmed again that 1,3,5 boxes for wages do NOT include HSA contributions or payroll deductions for healthcare line items.

I failed to notice that you said that you also made a personal contribution of $6,637.  That means that your deduction on Form 8889 line 13 would be $7,900 (limited to the maximum permissible) and you would have a $674 excess contribution on Form 5329.

I'm questioning whether or not you actually made a $6,637 contribution in addition to the $1,937 that was made through your employer.  Did you actually make only a $6,637 - $1,937 = $4,700 contribution separately from the $1,937 contribution made through your employer?

[Edit]  Also, given that you reported total contributions to a single HSA of $8,574, resulting in an excess contribution of $674, that fully explains the $202 increase in federal taxes due when you add the $1,937 entry.  That increase is 6% * $674 = $40 of excess contribution penalty calculated on Form 5329 and additional income tax of 24% * $674 = $162.