When you have received claim packs/renewals, presumably it states on there you are a single claimant?
It is your responsibility to check all information is correct and to inform tax credits of any errors (having only you on the form and not your partner who lives there would be an error).
If you have failed to calle tax credits to notify them of an error on paperwork then you will be liable.
It sounds like you called them initially and informed them (if you can prove this via call transcripts) then you may not be liable from this period until you receive any paperwork, as you have informed them of a change and they have failed to act on it.
However, once they send you paperwork, it is your responsibility to check all the information on there is correct.
You need to make a timeline,
So partner moved in 01.01.2016
Called tax credits 02.01.2016 to inform them of a change in circumstances. No action taken by tax credits.
01.04.2016 received renewals pack, states me as a single claimant, with 4 DC, earning X amount. Failed to inform tax credits that this was incorrect.
So from Jan-April you may be able to not pack pack if you can prove overpayments were their error but once you receive any paperwork from them, it is your responsibility to check all information on there is correct.