AV and TV mode are inverse of each other in this case. So you can use either one to accomplish your goal. It often comes down to preference but I use TV mode significantly more in this case.
1) I shoot a lot of tennis tournaments and do so most often in TV mode. Because maintaining higher shutter speeds is the priority and I desire full control over that variable. I can tweak exposure as desired with ISO...
2) In theory, AV mode would seem best for such because it will always render the fastest Shutter Speed possible for a given shot. There are however many challenges and issues with that approach.
a) What if you desire a slower shutter speed for a particular shot in depicting motion?
b) What if it is a dark day (morning, evening, cloudy) or your subject is dark or in and out of shady areas? In that case you will not maintain a higher shutter speed as desired. This is the issue the OP experienced, btw.
c) What if you need a higher Aperture setting in getting two people (doubles players) in focus? Once more, your camera will adjust your shutter speed to a slower setting in accommodating for that, which may be slower than you desire.
AV mode is perfectly fine if the lighting is sufficient to maintain high shutter speeds and such is what you desire for a given shot.
3) If the lighting is really challenging or I need greater control, I will resort to full Manual.
You may get a lot of debate on this but this is what has worked best for me. I play and photograph a lot of amateur tournaments where there are often great challenges (tree's, backgrounds, daylight hours, clothing etc).
Again, it really is what works best for you OP but I would recommend using TV mode and set the ISO as needed. That scenario requires less adjusting btw as you really don't have to adjust the ISO much. Just set it to around ISO 500 (typical) and raise it as light diminishes or is poor (morning, evening, shaded). If I have to go over ISO 1000 due to poor lighting I will then resort to Manual mode or a faster lens.
You should also consider your metering mode OP. I most often use center weight but will resort to spot metering if needed.