Do I need a new circuit breaker?
There may be a perfectly reasonable reason for the Pushmatic panel tripping with 2 major appliances going at once; however, that would also be a reason to replace the panel.  

Pushmatic is a fine panel

It does not have the problems of some old panels of being unsafe.  However, it does have a reputation for the action of resetting the breaker being very stiff, and of tripping too soon (i.e. Failing safe). These are convenience, not safety issues. 

A new 200A service

If you want a new 200A service and panel, you will need a new panel obviously, and an new meter pan, and a new service entrance. 

If I were you, first I'd go with a BIG panel (because 42 spaces vs 16 spaces is a tiny amount of money compared to total project cost).  And install it right next to the Pushmatic panel.  The meter pan goes outside, right next to the existing meter pan and pole. 

Then, I'd make the Pushmatic panel a sub-panel.  Install a 100A breaker in the main panel, then feed the Pushmatic panel off that breaker. That way you have very little disruption to the existing system wiring, and fairly low installation costs. 

At some point, the power company comes out and moves their drop from the old service pole to the new service pole.  Simultaneously, your electrician cuts the Pushmatic panel over to be a subpanel.  Your house downtime would be a couple hours. 

Once this is set up and you are happy, you can then move circuits over to the new service panel, one circuit at a time as convenient.  Or you don't have to. For instance I would move that dryer over first.  

If the Pushmatic panel has one main breaker

Then if you don't want a new panel right now, replace that main breaker.  

If you do want a new main panel, then bypass that faulty main breaker, and feed the main buses directly.  

If the Pushmatic panel has up to six main breakers

There's an obsolete style of panel called a "Split-bus" or "Rule of Six".  The Rule of Six literally means six hand motions to shut off all the main breakers. I.E there are multiple main breakers.  Some of them feed major appliances, and one feeds another section of the panel - a "subpanel within the panel" if you will.  This was called the "Lighting section" because lighting was the most common outlet load. 

This was done because a 100A breaker was prohibitively expensive, and a 60A lighting breaker + 30A dryer + 30A water heater + 50A range + 30A A/C is unlikely to peak all at once.  But the possibility of that is why we consider this panel style dangerous and due for replacement. 

However, this creates a possible reason for a breaker trip: If your dryer is punched down in the "Lighting Section", then it's sharing a limited 40-60A "Lighting" breaker with your washer and whatever other loads you have in the house. That would explain it! 

Feeding a Rule of Six panel from your new 200A panel

If you follow my advice of putting a new panel right next to this one, again, you do not need to replace the old panel. You can simply rig it up as a subpanel, or to be more precise, two subpanels.  

You feed the "main" section from one breaker, and the "lighting" section from another breaker.  Now you are putting 60A (or more if panel permits) into the lighting section, and 100A (or more) into the "main breakers" section, which is no longer that.  

The only issue is that you must also split the neutrals, so I would identify the circuits from the "main breakers" section that need neutral, and feed that separately e.g. Through a Polaris splice.  

Again, this results in very little disruption to the old panel.