My HSA contributions were $6300 for a family HDHP. turbo tax says I over contributed $5000 even though the limit is $8300. Is this incorrect?

Did you have a spouse with an HSA?  The $8,300 limit is a combined limit, not per spouse.  

I recommend you return to the HSA section of the program and check all your entries and answers.  To navigate to the HSA section, click on Tools, type in "HSA" and click on "GO".  If your $6,300 contributions was made through payroll deductions, be sure you don't enter any additional contribution amounts.  Assuming you had coverage under a High Deductible Health  Plan (HDHP) all year, be sure to indicate that, and also indicate that you were covered by a Family plan every month of the year.  

@smcpeterson16 here is a common error:  when the HSA contributions are withheld from your paycheck (code W in Box 12), do NOT replicate that later on when TT asks if you made contributions to your HSA. 

When it is asking if you made contributions to your HSA, you are NOT to include what was withheld from your paycheck so this entry may be zero.  

I have left that box saying $0 when asked if I made additional contributions. It is double counting the box from box 12 it seems since my total was $6634. 6634X2=$13628 ; 13628-8300=$4968. TT is saying i owe the $4968. I have not double entered this anywhere and am not sure how to correct this error. @NCperson @DavidD66 

@NCperson  @DavidD66     Also, My spouse does not have an HSA. 

Somehow you've created a double entry.  You might consider deleting all W-2s, next deleting Forms 8889, then reentering the W-2s and redoing your entries in the HSA section, making sure to leave blank the box for entering personal contributions.