I bought (and reserved seats) online for 2 passengers to travel from Kings Cross to York (and return) on 16th March a few weeks ago. A week later, the RMT announced a new round of strike dates - including the 16th March
What are my options please?
As a basic entitlement, regardless of strikes, you can always change an Advance ticket to another date or time by paying any difference in fare, plus any admin fee charged by the retailer in question.
If (as is likely to be the case) LNER advise against travelling on 16 March, you can obtain a full refund of your tickets if you decide no longer to travel. This applies even if your train is not cancelled. It is also likely that you will be allowed to use your ticket (without having to make any changes to it) on alternative services that meet your ticket's operator restrictions, either the day before, or up to several days after the strike.
If your train ends up being cancelled or retimed as a result of the strike, then above and beyond these rights, you are also entitled to take the next service complying with your ticket's operator restrictions.
If you are likely to arrive more than an hour late at your final destination by waiting for the next train that meets these restrictions, then your rights increase further. You become entitled to require LNER to re-route you at the earliest opportunity, including on other operators such as Grand Central if relevant. Alternatively, you can choose to have your ticket re-issued for another date of your convenience (which could be beyond the couple of days where there is general ticket acceptance) without paying anything extra.
You can of course decide to still travel on 16 March - there will be services, albeit running to a significantly reduced timetable, likely with first departures no earlier than 07:30 and last arrivals no later than 18:30.
Really it therefore depends on what you would like to do. You can travel on that day, or (likely) travel on another day or get a refund.