If I understand you correctly, the two photo shoots (with no catalog backup) have been imported in Lr on your old computer and are therefore in the catalog on your old computer but you cannot access it any more. Therefore you are thinking of loading your latest catalog backup onto the new computer - but this catalog backup does not contain the two shoots.
If my assumption is correct you should - after installing Lr on the new machine and loading the latest catalog backup - import the photos from the two shoots. Since they are not in the loaded (backup) catalog, Lr will treat them as new images. Lr will read the xmp-file on import and display your images accordingly. But you won't see your previous editing steps in the History panel.
Make sure that in the Import Dialog on the right side under <Apply During Import> you select <None> for <Develop Settings> and for <Metadata>.
Naturally it would be better to transfer your working catalog from the old computer to an external hard drive and from there to your new computer. You won't need Lr for that. Just copy or move the file with the ending < .lrcat> to the external drive via Mac Finder / Win Explorer. Do a search for files ending with < .lrcat> and - if you find several - transfer them all. You can sort out later which ones you won't need anymore.
If you manage to salvage the catalog from the old computer open Lr with this catalog by double-clicking it. You only will have to do the re-linking of your folders in case they are on drives with different drive letters. It will make it easier if you can give your drives (including the external ones) the same drive letters they had with the old computer. The catalog contains also your history, so you would see the history steps for the photos of the two shoots.