I was really meaning to write to you as soon as I got here, but I was very busy sightseeing, as you can imagine.
I was really meaning equates with I was going to, so they are basically synonyms. So if you had the option to write I was going to write to you as soon... you'd see that it does sound natural. As soon as I got here, is just sequence of tenses because the first verb was in the past tense, this verb has to be in it as well. I was very busy is obvious, it is a continuous tense in the past.