Yes,
When you say 'we' I assume that it is you and a travelling partner- get yourselves a two together railcard at £30 for the two of you, rather than a senior railcard at £30 each.
https://www.twotogether-railcard.co.uk/
Irrespective of train operating company you can buy all your tickets from one company- Americans seem to find the LNER site to be most user friendly.
https://www.lner.co.uk/
Buy your tickets in advance for the best fares.
At the moment there are no through advance fares to Moreton from anywhere due to the bridge reconstruction south of Oxford.
So on Sunday 14 May- you will need to travel from London Marylebone (NOT Paddington) to Oxford by Chiltern trains- at 42 minutes past each hour to Oxford arrive at 8 minutes past the hour (1 1/2 hours later), change trains, depart Oxford at 50 minutes past each hour.
There are no advance fares currently. It is cheapest to buy tickets on the day on this route-
2 Together for 2 people London Marylebone to Oxford £41.80, then buy separate tickets from Oxford to Moreton 2T for 3 people £16.60.
This is NOT the normal situation, where advance fares at a fraction of this price would be available.
You could travel on the 12 minutes past each hour train to Oxford, but the hourly connection at Oxford is tighter.
For 17 May, due to the lack of through fares buy this as
2T for 2 People Moreton to Worcestershire Parkway-
on the 0918 train is £32
on the 1019 train and all later trains is £21.10
Then buy separate advance fares Worcestershire Parkway to York, on the 1014 and 1114 trains only is £62.10 for 2 people on 2T.
You have 25 minutes to change from the High Level platform to the Low Level Platform at W Parkway. So your best value train is the 1019 train from Moreton then the 1114 train from Worcestershire Parkway. You will have a change of trains at Birmingham New Street.
On 21 May Edinburgh to York buy an advance train- 2T for two people that day on many trains is £44.30. Most of these are LNER trains, the general view is that their trains are better than Cross Country anyway.
The best thing to do is to buy your tickets first (specifying that you are on a Two Together Railcard), then buy your Railcard at any time before you arrive in the UK, or even at your first station after arriving here.
To get the advance fares to York and Edinburgh book ASAP. The prices will only go up.