If I have a misdemeanor ticket that I haven’t paid off and now I have a new one for a separate incident, can I get arrested if I show up at my new court date?

If the court has issued a capias or warrant because of violations with your other case, then you could be arrested if you show up to court.  If the judge has not issued a warrant, then he usually won't become motivated to do so until someone brings it to his attention.... so everything really depends on what your judge knows and how he may or may not react to your situation. 
The good thing is that you know about the offense and they are still basically tickets.  You have a couple of different options.  The first is to consult with an attorney that routinely handles these type of tickets.  Even if you don't hire the attorney, they can give you some basic pointers on your judge and the prosecutor that you'll be dealing with.  Many attorneys have what they call "ticket mills."  Basically, these attorneys only handle tickets or ticket offense that can be run quickly through the system.  They tend to be significantly cheaper than the plain jane defense attorney.  If you simply cannot hire an attorney... still do the consults and then ask to talk to the prosecutor when you go to court to see if they will work with you.  When you visit with the prosecutor, only discuss your plea options.  If you admit or confess to the offense during the negotations, then you will potentially hear the statements later at trial... so keep mum on offense details and only explore sentencing options.  I know you don't want to serve jail time, but without sufficient resources, this may be your only option.  If you have to serve some time, see if you can either do your time concurrent with all of your other cases and request a credit for your time against the outstanding tickets.  Basically, you get double the bang for your buck.  Another option is work release.  More jurisdictions are offering this as an alternative to reduce jail over crowing issues.  Basically, you spend your time in jail... except that you are allowed to go to work as long as you report back to the jail at the end of your shift. 
None of the options are perfectly ideal, but each has a the potential to resolve all of your warrant issues.