The answer to your question will vary widely based on where you are and what kind of campaign it is. However, there are some important considerations that will apply in most cases.
First, the level of coordination you hint at in your question (sharing staff, etc.) may not be permitted. You should be certain you understand campaign finance law in your state, and FEC rules if it's a campaign for federal office. Specifically, it's very likely you can't absorb local party resources without counting them as donations (perhaps in-kind donations), and those donations will be subject to limitations. This problem can be avoided by paying (fair market value!) the local party for these resources.
Next, whether you coordinate with a local party will be governed at least in part by the degree to which it's organized. Where I've worked, counties are among the most important levels of party organization - and some county parties are very well organized while other county parties barely exist. You need to identify what unit of the party you should coordinate with, if you're going to at all, and then figure out how well organized it is. If it's poorly organized it probably won't be helpful to you - though if you share broad goals perhaps your campaign could help to organize it better, around your campaign this year, but continuing into the future.
So suppose your local party is well organized, both sides want to coordinate, and you figure out any campaign finance issues. What might the relationship look like?
Are you the nominee yet? Some states have had their primaries and some haven't; if your primary hasn't happened yet, you may or may not have opposition. If you're pre-primary and you have opposition, it might be very awkward to have any official or extensive relationship with the local party. For the most part, party organizations don't see themselves as arbiters of primaries, but as supporting structures for winning the general election. So, if you approach the party before the primary and they give you the cold shoulder, win the primary and try again.
Once you're the nominee, a campaign's relationship with a well-organized party might be very helpful to both sides. The campaign could rent space, which can usually accommodate whatever arrangement the campaign wants (good for you) and provides rental income for the party (good for the party). You could rent a room, half the office, or just the right to have a meeting occasionally. (The party may not have a permanent space of this nature - see above, how well organized in the local party?) Party staff or volunteers could take messages for the campaign, they could store yard signs, they could provide a venue for media availability, etc. etc.
Perhaps most importantly, by being the nexus of local campaign coordination, the party can help different campaigns connect and coordinate among themselves. Your volunteers might not mind talking to people about another campaign or two; another campaign's volunteers might not mind dropping lit for you, setting up your signs, mentioning you on calls, etc. In my experience this is difficult to establish, but beneficial to all sides when it works out.
I want to say something about the state party. Any well-organized local party will be a subsidiary part of the state party, and the state party might be much, MUCH more helpful to your campaign. Again, this depends a lot on what you're running for. It also depends on how professionalized your state party is - this varies a lot across states, but most have pretty professional organizations at the state level. The state party might hold a unique position in state law, one that might allow it to help your campaign financially - sometimes very significantly. This is probably only important if you're running for state legislature or above.
Finally, you say
candidates can run under a party's name with little or no support or
recognition from that party.
...sort of. It depends on how you conceive of the party. Consider closed primary states; one doesn't need to be a member of the party in order to vote in the primary, but one DOES have to be registered as that party to vote. Does that make each voter part of the party? I'm tempted to say yes, but it's a contestable position obviously. In open primary states it's even harder to say that, but when you vote in a primary you're participating in an analogue to a convention, just with many fewer requirements. That said, you can't participate in more than one party's primary, certainly in the same election. Anyway, I don't think the American party system is as weak as you make it out to be.
[This is based on maybe a decade of campaign experience, mostly in the Midwest, and several years of graduate study in political science.]