DVLA V5C
As regards the Traffic Enforcement Centre, they're more interested in how quickly you applied to change your V5C after moving. When the V5C was issued is a clue, but it's not the answer. If you applied by post and DVLA took a week on receipt then it's not the gap of 14/5 to 14/6, it could be 15/5 to 16/5, the rest of the period 16/5 to 14/6 being down to DVLA, not you. I've given an extreme example, but the point remains.
Delivery of PCN
Although the authority has to use the RK address at date of contravention, whether you were 'at home' to receive post is contingent upon when the PCN was sent. They have 28 days. As above, the PCN might not have been issued until 10/6, in which case a gap of 10/6 to 14/6 would be a shoo-in with TEC.
We need more essential details pl.
So, IMO:
Pay CDER, as this caps your liability;
Find out the date the PCN was issued;
Tell us how and when you notified DVLA to change your V5C.