What were your previous bikes and did they have the larger diameter 700C tires on them? What model of bike is this one? Perhaps what you are feeling as insecure is just getting comfortable with a larger wheel and larger performance oriented bike frame that might actually be feeding you the feeling of the road more.

However do check your tire pressure often. If it's been a week since you last checked, then it's well past time to check. On my narrower 25mm tires I check them and put air in them every 2nd or 3rd day. The other thing to check is whether all the spokes feel about the same tension to your hand/fingers and they aren't too loose altogether.

Don't know about your area, but bike shops here are pretty much owned or managed by the mechanic. And they all are willing to look at your bike and give you their assessment of what it needs for free. Whether you leave the bike with them or take it home to do the fix yourself is okay with them.