The statement of truth is not crucial. However it helps to know what LBBD have said, so you can rebut it. When you file an OOT application the Enforcement Authority should send you a copy of why they oppose the application, if they oppose. LBBD seem to believe that such rules don't apply to them so they frequently don't bother.
It would also be good to have some evidence that you applied for the V5c logbook address update before the 8th of April 2022. Otherwise the District Judge might just conclude that you didn't apply until May 2022, so this is a problem of your own making,
However the Royal Mail redirect is also good evidence. Even if you didn't actually apply to update your address until after April the 8th, you should still have got the notices in the post via the re-direct.
You could also try contacting
Bailiff Advice Online. They may be able to help you complete the N244 applications in return for a small donation.
Try to keep the case very simple.
On date X I changed address from A to B
9th Mar 2022 I paid for a Royal Mail redirect for three months .
On March ?? 2022 I completed the change of address form for the V5c logbook and sent it to the DVLA.
8th April 2022 is the date of the two alleged contraventions.
The Borough requested the Registered Keeper details from the DVLA but though no fault of mine the address update had not yet been processed. Delays with the DVLA were widespread news in 2021 and 2022.
etc.
etc.
As a result I did not receive the initial Penalty Charge Notices and was denied my right to challenge the Notices and to appeal to the Independent Adjudicator.