If you were born on the day the sun changed signs, you aren’t some of both. It really doesn’t work that way.
The sun is a distinctive energy in the chart. The signs and degrees are as well. Having a planet at 0 degrees of any sign is very different than 29 degrees.
If you were born on a “cusp” day and you feel some connection to the two signs it’s most likely because you have some planets in both signs.
Mercury is so close to the sun (no farther than 28 degrees), it is always in either the same sign as the sun or a sign adjacent to it. For example, if you have the sun in Cancer, your Mercury is in Gemini, Cancer or Leo.
Venus is also really close (never farther than 48 degrees) so it can never be more than two signs away from the sun. If you have the sun in Leo, your Mercury is in Gemini, Cancer, Leo, Virgo, Libra.
Lots of people have Mercury and Venus in a sign right next to their sun sign.
Then there’s all the other planets, plus the nodes, so having something in a sign adjacent to your sun sign is not unusual. As you learn about house placement and aspects, you start to get a feel for the depth and complexity of each piece in your chart.
If you feel a connection to a sign adjacent to your sun sign, there is a reason for it in your natal chart and it has nothing to do with cusps.
Most people know their sun sign by the day they were born. If you were born around a sun sign change, then it means you need to cast your natal chart to know your your sun sign. There are lots of free sites that do this for you online. Just search “cast natal chart” and you’ll get a long list. You’ll need to enter your date, time and location of birth and then the chart is cast your you.
If all you’ve ever known is your sun sign, cast your natal chart and start reading about your moon sign or some of the other planets.