Wow I'm really surprised no one provided an answer. Honestly, the following information would be helpful if you want any neat situation specific advice. Vehicle type model, year and miles, example 2001 Toyota Echo 4 door 157,000 miles. I realizes that information isn't what you asked, but sometimes you get cool free advice from people.
To answer your question, yes i shop around. I don't think its frowned upon, but I don't care anyway. Sometimes it seems people quote vastly different quotes for the same job, I think some mechanics don't like getting into uncomfortable positions and purposely quote prices to steer you away if the job requires awkward positions. I may not be the best person to ask though because I usually DIY everything, but if the weather outside is extreme I will shop around for quotes for some jobs. Also I usually only get verbal quotes if that helps ( I haven't been screwed over....yet). But in your situation it makes sense (to me) to drive to the locations. Sometimes you end up getting valuable information for free, while they attempt to quote a repair price.
In regards to your car, I honestly find it strange your primary mechanic wont do the A/C. If it really is just a compressor change that should be well within his comfort zone if hes been in business for a while. Also if your DIY comfortable and "enjoy" the challenge I say go for it. If you found a youtube video the hard part is 50% done lol. IF IT WERE ME, I would DIY the actuator part, as far as the compressor goes, I might DIY it, or I might shop around. Just depends on the weather. I dont work well under a car if the temperature isn't between 60-80degrees out. That's just me, I'm VERY good with cars so your situation and comfort may vary.
Again my opinion is it makes sense to get different quotes for a specific job even if it does take some time, especially since mechanic quotes can vary by 100s of dollars.
Hope my rambling helped in some way.