The construction with would is probably the one most frequently used here, and that with will next, but the others are all acceptable. At a guess I'd say was is probably the least frequently used, followed by am and will be. That reflects the relative infrequency of simple present in non-backshifted use.
It is not entirely irrelevant that all of these except was can be contracted, and probably would be in conversation.
He invited me for dinner, but I told him I'm out of town tomorrow.
He invited me for dinner, but I told him I'll be out of town tomorrow.
He invited me for dinner, but I told him I'd be out of town on tomorrow.
Among other things, that has the effect of backgrounding the verb and throwing the focus onto out of town. In conversation, nobody's going to notice the difference.
FumbleFingers points out that there's another option, constructions with BE going to be:
... I told him I was going to be out of town tomorrow.
... I told him I'm going to be out of town tomorrow.
These two are, I'd guess, on the upper end of the frequency scale.
FumbleFingers also points out that I was is often elided in speech. He implies that the British elision is /aɪz/; in US speech it is /'aɪ(j)əz/ or (in my dialect) /'aəz/. But this is not an "officially approved" contraction; when it's spelled out in writing (typically as "I uz") it is an "eye dialect" form which imputes illiteracy to the speaker.