This is really not practical. A taxi between the two airports would probably run $80 to $100 each way (JFK to Manhattan is $60 with tip) and you would have to allow at least 1.5 hours each way. by publick tansport, you could do it for about $20 each way (maybe a bit less) but need to add at least 1 more hour if not more in case of delays. Plus the left items will cost $4 or more a day so if your trip is 2 wks, you would be spending $100 ti $300 or more to leave a couple coats. I travel often in the winter to tropical places. I NEVER take a coat or even real shoes. You really aren't outdoors for that long getting to/from the airports, even if you take public transport. Wear a sweater to the airport - you will probably want it in the plane anyway. If you are taking only sandals, wear warm socks with them on the way to/from the airport. Might not look fashionable but it does the trick, saves you money and headaches. Also less luggage to worry about luggage fees.
Leaving Luggage at JFK returning to EWR