You should be fine. While it's technically true that Likert Scale data is ordinal categories, it is often assumed to be interval as long as:
1. The intervals between the values are generally equal: is the difference between a 1 and a 2 the same as between a 2 and a 3?
2. The underlying concept you are measuring is itself continuous.
The sum of ordinal variables is still ordinal, so running them separately instead of summing them doesn't help anything.
You have a few options. Run the paired t-test as planned, and check your assumptions well. If they look good, stick with the t-test.
If the assumptions don't turn out well, there is a non-parametric equivalent to the paired t-test called the Wilcoxan sign-rank test. It is available in SPSS under nonparametrics.