Were there professional musicians in the era of Romantic music?

There certainly were. By that time you find a lot of similarities with today's classical musicians. You find virtuosi on tours, playing concerti with orchestras or in solo recitals. You find great singers in recitals or singing roles in operas. You find composers and conductors. And you also find orchestral musicians, in orchestras very similar to ours.

what were their lives like? Was it possible to make a living playing music during this time period?

Yes. There have been professional musicians since forever. The status of the musician has changed. At some points they have been low in the social hierarchy, but by the 19th century many musicians were very well respected.

Not all musicians lived the same kind of life. Paganini and Liszt were rockstars, but there were plenty of musicians trying to make ends meet: looking for gigs playing in churches or being invited to extras for an orchestral concert, opera, musical theatre, etc. They would also sometimes have students.

I think you should really take a look at Berlioz's Evenings with the orchestra (Soirées de l'Orchestre). You can find it for free in the Gutenberg project (only in French), and you can easily get it in English.

You can also read the correspondence of several famous composers to get a closer look at their lives (Chopin, Tchaikovsky, Berlioz...). There are biographies from the period (Liszt wrote one of Chopin, his buddy), and some autobiographies (for example Berlioz's).

I am into the concert side of music, but you can obviously also find professional musicians of other types in this period.

Well, there have been professional musicians in European music (which your question seems to be about, as you talk about the Romantic period) for very, very long. Just think about the troubadours!

My old trumpet teacher used to say trumpetists during the 18th century were extremely well payed due to the skill needed to play melodies on the pre-valve trumpets (you would need to get very high up in the harmonic series before you could play chromatically, and that took some extreme muscle work). As a result, the trumpet players were always filthy drunk, as they could pay for it. This is probably a big fat myth, but there could lie some truth in that trumpet players were better payed, but we need someone to source that before assuming so.

I'm not a historian, just a composer, so I don't have very good sources. However, I would recommend that you research composers like Chopin, Paganini, Clara Schumann and Scarlatti (last one 18th century, but still) who were not only composers but frequent performers. Since it seems we mainly document the music creators life, I would presume you would find some answers to your question by looking into them.

And we shouldn't forget singers, who are of course also musicians. For example, take famous opera singers like Jenny Lind who very much indeed could pay her bills and then some.

I would be interested in some history about the not so famous everyday musicians at this time though. How much did you have to travel to get a job as a musician, were there manageable ways to live outside of the court, operas and capitals?