Can you still contract a disease after being vaccinated and be able to spread it?
I tried to break down your question into smaller parts and answer them individually:
Q. What is immunity?
A. Simply put, immunity is the ability to react to an infectious agent and stop it from causing disease.
Q. Do vaccines make you immune to infectious diseases?
A. Yes, by definition. If it doesn't make you immune, it's not a vaccine.
Q. Can we get disease even after being vaccinated?
A. Yes, because no vaccine is 100% efficacious. For example, the MMR vaccine has about 97% efficacy in preventing measles.
Q. Do vaccines prevent infection?
A. It depends on how the vaccine works. Some vaccines enable the immune system to actually reduce survival and multiplication of the infectious agent. The polio vaccine is an example$^1$. It leads to generation of antibodies that neutralise the virus$^2$.
Other vaccines prevent disease (or reduce its severity) but do not prevent infection. The diphtheria toxoid is a good example of this$^3$, because the antibodies generated by vaccination neutralise the toxin while leaving the bacterium unscathed. (I believe all toxoids behave like this.)
Q. Can we spread infection even after getting vaccinated?
A. Yes. This is easy to see for those vaccines (like diphtheria, mentioned above) that prevent disease but not infection.
Vaccinated individuals can spread infection with the other, multiplication-stopping kind of vaccines too. Again, this happens because no vaccine is 100% efficacious. Note that chances of spread are reduced (as compared to the unvaccinated scenario), even if they are not zero. For instance, see the work of Seward et al$^4$ on breakthrough varicella.
Q. The big question: is getting vaccinated more dangerous than not getting vaccinated?
A. Sorry, but this is not the place to discuss it. It requires a careful look at the risks and benefits of vaccination, not just for the individual but also for the community at large. That would be a discussion more appropriate for Medical Sciences Stack Exchange.
To summarise: yes, we can catch a disease and spread it even after being vaccinated. The chances of this happening vary with the kind of vaccine.

Journal articles cited

Howe HA. Am J Public Health Nations Health. 1957 Jul;47(7):871–5. https://doi.org/10.2105/ajph.47.7.871
Faden H, Modlin JF, Thoms ML, et al. J Infect Dis. 1990 Dec 1;162(6):1291–7. https://doi.org/10.1093/infdis/162.6.1291
Miller LW, Older JJ, Drake J, et al. Am J Dis Child. 1972 Mar;123(3):197–9. https://doi.org/10.1001/archpedi.1972.02110090067004
Seward JF, Zhang JX, Maupin TJ, et al. JAMA. 2004 Aug 11;292(6):704–8. https://doi.org/10.1001/jama.292.6.704

Suggested reading

https://www.who.int/health-topics/vaccines-and-immunization
https://www.ncbi.nlm.nih.gov/books/NBK27131/