They are considered exotics

Shoot both no doubt. If you don't, someone else will.

I have seen many on my low fence ranch. They come and go.
I would shoot only the bull elks.
State biologist estimates a free range group of a few hundred in the Edwards plateau area to over a thousand free range across Texas. I usually see them in groups from one up to five on my place.
I have let several cows go when hunting especially if with a calf.
It is fun to see them. But this is my thought as a land owner. I find myself being more conservative on harvesting animals after becoming a land owner.

They are exotics in Texas so you can hunt them year round and even at night.

I had to put a timer on my protein feeder. I had a group of five elk wipe out 2,000 pounds of protein in a single weekend.

I'd hang your feeder on a tripod to eliminate them emptying it in a weeks time, then wait and see how many you really have there.

Elk steaks are choice I'd admit!

If you whistle they will probably come to you. Don't shoot pets