You need to know the exact size of your screen and make sure whatever projector you want to buy will work from the throw distance properly and work. If you don't know your exact screen size and throw distance, then you won't get the best responses.
If you want a good image and you have a larger screen, then the W7000 from BenQ is the one I would recommend. It is bright enough to give a nice vibrant image with good color, and the use of DLP technology handles motion a bit better than LCD projectors do. The JVC LCoS projectors are great for movies, but aren't as bright so about 110" is as large as I would recommend for their projectors right now.
There are no $1,000 projectors which will really shine at the size you are at, or will shine in general. They are cheap bottom of the barrel stuff. They look alright, even good, but have significant limitations with brightness, color accuracy, motion handling, black levels, and scaling which just makes me recommend you avoid them.
The better 2d 1080p entry level projectors just don't do a great job with detailed motion like you need while viewing golf.
If you provide detailed height/width measurements and a exact screen to lens measurement, then I could more accurately recommend a projector, but the reality is that $3,000ish is the sweet spot for a mid-line 1080p projector. Your screen in there sounds like it is FAR larger than mid-line of 92" to 120" diagonal.