No, the limit on annual contributions is based on the number of months in which you were covered (except for the last month rule, see below).
If you dropped out of the HSA eligibility during the year, then on the screen entitled "Were you enrolled in a High Deductible Health Plan (HDHP) in 2016?" you would answer "no" to the question "I was enrolled in an HDHP as of December 1, 2016".
This will trigger a new screen that will ask you to indicate which months you had the HDHP.
The limit that is computed is generally going to be 10/12ths of the $3,350 full limit.
When you enter this, TurboTax will determine that your $3,350 contribution in the spring was over the limit, as adjusted, and tell you that you have an excess contribution.
TurboTax will then ask if you will withdraw the excess contribution before April 18, 2017; if you say "yes", then there will be no penalty for the excess contribution.
Note that you must call your HSA plan administrator and tell them of the excess contribution and that you would like to withdraw it.
***last month rule***
As for the last month rule, it's a special rule that allows you a full year's annual limit, if you are in the HSA on December 1, 2016. This is for people who enter the HSA during the year - they get the annual full limit if they invoke this rule.
However, since you weren't in the HSA on December 1, 2016, this rule does not apply to you.
