A quart every 6-700 miles is getting close to being a problem.
You will lose a certain amount of oil from volatility, more-so if you are running hot. In the NOACK Volatility test, AMSOIL 10W-40 loses 4.86%, and others range on up to Valvoline 12.18%. Depending on what you are using now, Mobil 1 could well give you reduced consumption.

Before changing to synthetic, try using a good engine flush. Best quick ones I know of are made by AMSOIL and BG. For a slower flush (couple of thousand miles) look at Auto-Rx. Haven't used it myself, but many good reports.

If you have a mechanical problem, bad seal, bad gasket--not likely that any oil is going to fix the problem.

Mobil 1 is compatible with the non-synthetic. Putting a quart in as makeup will start a cleaning process. However, you'll get better results by making a complete change. If you don't, or won't, flush, watch the oil level. If you see a rather sudden increase in consumption after a couple thousand miles, change the filter and top off.