Yeah I was going to say 1060 3GB, but didn't know whether he could afford a 6GB version of it. The 3GB one is OK for most games at 1080p, but there are some games now where you'll not be able to use max texture settings due to the 3GB VRAM.
Going forward that number of games will only increase. For this reason I suggest you wait to see what holiday pricing will be on the 1060 6GB and try to add another 1-3 month's savings as well.
If you're like most gamers that are on a budget, and everything you've said indicates that, you will likely be on this GPU 3 or more years. That's a LONG time to be increasingly limited by VRAM demanding games.
Bottom line, weigh this decision carefully, it's an important one for future readiness. Where major components are concerned, the impatient buyer is often the regretful one. This is why I held out to see what your MAX budget was.
Truth is your budget is not far off from a 6GB model. The mini one fan ones start at $250, and the dual fan ones $270, and that's without holiday pricing. One of the best performers at a pretty good price with solid reviews is this EVGA. It's $255 through today.
https://www.newegg.com/Product/Product.aspx?Item=N82E16814487261&ignorebbr=1&nm_mc=AFC-C8Junction&cm_mmc=AFC-C8Junction-PCPartPicker,%20LLC-_-na-_-na-_-na&cm_sp=&AID=10446076&PID=3938566&SID=